Notion, en rapport avec la théorie des ensembles et des infinis mathématiques, et notion, en rapport avec la notion de cardinal d'un ensemble et en particulier, en rapport avec la notion de cardinal d'un ensemble infini ou de cardinal infini d'un ensemble.
Remarque : Les fichiers sur fichier-pdf.fr qui ont un statut privé sont bel et bien accessibles, qu'on en soit le propriétaire ou non, et ce en ayant la connaissance de leurs liens et en créant un compte : Il faut laisser ouverte la page initiale où sont listés les liens des fichiers ayant un statut privé et/ou y revenir après avoir créé ou ouvert un compte, tout en maintenant ce dernier ouvert.
NB : 02-11-2023 : Depuis peu, la table des matières n'est plus (accessible) dans le corps des travaux de recherche. On ne peut y accéder qu'en allant sur (la) Wikiversité à l'adresse suivante :
- le lien donné à la fin de la présente version de mes travaux sur le cardinal quantitatif, lorsque celle-ci est en PDF, pour obtenir la table des matières de cette présente version
- ou bien "Recherche:Cardinal quantitatif (table des matières, simplifiée)" , pour obtenir la table des matières actualisée de mes travaux sur le cardinal quantitatif
et en cliquant sur le bon icône.
(L'auteur n'est pas responsable de cette situation.)
NB : Les formules en LaTeX présentes dans la table des matières ne s'affichent plus correctement, depuis novembre 2021.
(L'auteur n'est pas responsable de cette situation.)
Version PDF de 213 pages et de 1,1 Mo de mes travaux sur la F-quantité (anciennement le cardinal quantitatif) du 17-09-2025, avec la table des matières qui s'affiche correctement et avec la numérotation des sections https://www.fichier-pdf.fr )
Depuis un bon moment avant le 14-09-2025, il n'y a plus de numérotation des sections et des sous-sections, ce qui ne facilite pas le repérage et la navigation dans mes travaux.
(L'auteur n'est pas responsable de cette situation.)
J'ai l'impression que les codeurs de Wikiversité, à force d'obéir à certaines injonctions du monde numérique actuel, dégradent, de plus en plus, l'affichage voire les fonctionnalités des pages des travaux de recherche. De plus, après qu'on ait préédité un passage et qu'on l'ait prévisualisé, le curseur ne revient pas exactement là où il était, initialement, mais au début de la section ou de la sous-section concernée : Ce qui constitue une perte de temps pour moi.
NB : La version originale de la présente version de mes travaux sur le cardinal quantitatif, lorsque cette dernière est en PDF, est également accessible et disponible sur (la) Wikiversité, à partir du lien donné à la fin de cette dernière.
NB : Il arrive parfois lorsque je copie-colle des passages de mes travaux à certains endroits, que ceux-ci soient aussi copiés-collés, malgré moi, à d'autres endroits. Le plus souvent, je parviens à supprimer les doublons en question, mais il peut arriver qu'il en reste certains.
Certaines mises à jour et modifications impliquent d'autres mises à jour et d'autres modifications en chaîne, parfois délicates, pour lesquelles il m'est parfois difficile de {détecter|repérer} et de déterminer les endroits où je dois les faire et/ou qu'il m'est difficile de faire dans la foulée, compte tenue de la longueur du texte de mes travaux.
De fait, il peut (encore) rester quelques passages écrits incohérents ou contradictoires, mais sans que cela ait, nécessairement, de conséquences sur mes travaux.
Mises à part les discussions associées à mes travaux mathématiques sur la Wikiversité, vous pouvez aussi vous rendre sur mon forum pour en discuter et les critiquer de manière constructive, en tant qu'invité ou en tant que membre (mais il faudra alors créer un compte pour vous y loguer) :
Tous les liens et toutes les discussions à propos de ces travaux mathématiques sur les forums de mathématiques : "Les-mathematiques.net" et "Maths-Forum" sont désormais périmés et obsolètes. La présente version de mes travaux mathématiques qui est aussi celle qui fait foi, est la version actualisée de ces derniers. De plus, de nombreux commentaires qui sont relatifs à ces discussions ont été donnés dans la page de discussion associée à la présente page de recherche, ainsi que dans une partie des "Passages que l'on peut omettre" et sur mon forum.
Utilisateur:Guillaume FOUCART/Passages que l'on peut omettre/Passages dont on peut omettre certains passages, dans ma page de recherche principale/Commentaires, impressions voire spéculations autour des amateurs, des shtameurs, de moi-même, des intervenants et des grands intervenants sur les forums de mathématiques
Malgré le foisonnement de titres et de sous-titres : Avec une échelle réduite de 50%, les travaux, dont il est question, ne font que 56 pages, au format A4, le 29-03-2021, et encore ils sont, relativement, aérés et espacés. Certes, ils ont, trompeusement et faussement, l'allure et l'apparence d'un mille-feuilles argumentatif, mais, concernant la partie spéculative, ils sont, peut-être, attaquables, et s'ils le sont, ils peuvent, peut-être, être démontés et anéantis, uniquement, concernant 2 ou 3 points fondamentaux voire cruciaux, bien ciblés. En moyenne, chaque sous-partie élémentaire mentionnée dans la table des matières est relativement {courte|brève} : Il n'y a donc pas lieu d'être effrayé par le grand nombre de sous-parties élémentaires figurant dans la table des matières. Par ailleurs, il y a beaucoup d'exemples illustratifs.
VOICI LA TABLE DES MATIÈRES DÉTAILLÉE LE PLUS POSSIBLE (Il faut d'abord lire les titres en gras. J'aurais aimé pouvoir disposer d'une table des matières qui se déploie au fur et à mesure que l'on avance en allant des titres généraux aux titres particuliers. Il est très rare que les définitions, les propositions, les lemmes, les théorèmes, les remarques,
⋯
{\displaystyle \cdots }
[NB : Désormais, on peut aussi consulter la version de mes travaux, avec une table des matières, simplifiée (Cf. liens ci-dessus).]
Cardinal quantitatif (nouvellement, F-quantité) sur
R
n
{\displaystyle {\mathbb {R} }^{n}}
R
″
n
{\displaystyle {\mathbb {R} ''}^{n}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Remarque : L'introduction n'est qu'une petite partie de mes travaux : N'oubliez pas aussi d'aller jeter un coup d'œil sur le reste ou de le survoler ou de le consulter. Si dans l'introduction, il y a beaucoup de texte : Dans le reste, il y a beaucoup de formalisme et de formules mathématiques.
J'utiliserai une terminologie personnelle, en renommant parfois autrement certaines notions existantes.
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
PV (comme « petite variété ») les sous-variétés compactes, convexes (connexes) de
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
C
1
{\displaystyle {\mathcal {C}}^{1}}
et
PV2 (comme « petite variété 2 ») les sous-variétés fermées, non bornées, convexes (connexes) de
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
C
1
{\displaystyle {\mathcal {C}}^{1}}
et on posera :
P
V
(
R
n
)
=
{
A
∈
P
(
R
n
)
|
A
s
o
u
s
-
v
a
r
i
e
´
t
e
´
c
o
m
p
a
c
t
e
,
c
o
n
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e
x
e
(
c
o
n
n
e
x
e
)
d
e
R
n
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
o
u
s
a
n
s
b
o
r
d
}
{\displaystyle {PV}(\mathbb {R} ^{n})=\{A\in {\mathcal {P}}(\mathbb {R} ^{n})\,\,{\Big |}\,\,A\,\,sous{\mbox{-}}vari{\acute {e}}t{\acute {e}}\,\,compacte,\,\,convexe\,\,(connexe)\,\,de\,\,\mathbb {R} ^{n},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,ou\,\,sans\,\,bord\}}
P
V
2
(
R
n
)
=
{
A
∈
P
(
R
n
)
|
A
s
o
u
s
-
v
a
r
i
e
´
t
e
´
f
e
r
m
e
´
e
,
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n
b
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n
e
´
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
d
e
R
n
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
o
u
s
a
n
s
b
o
r
d
}
{\displaystyle {PV2}(\mathbb {R} ^{n})=\{A\in {\mathcal {P}}(\mathbb {R} ^{n})\,\,{\Big |}\,\,A\,\,sous{\mbox{-}}vari{\acute {e}}t{\acute {e}}\,\,ferm{\acute {e}}e,\,\,non\,\,born{\acute {e}}e,\,\,convexe\,\,(connexe)\,\,de\,\,\mathbb {R} ^{n},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,ou\,\,sans\,\,bord\}}
La notion de F-quantité est la {vraie|véritable} notion de nombre ou de quantité d'éléments d'un ensemble, qui est une notion au moins définie et construite sur
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
mesure
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
1
{\displaystyle 1}
i
(
0
≤
i
≤
n
)
{\displaystyle i\,\,(0\leq i\leq n)}
R
n
{\displaystyle \mathbb {R} ^{n}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
principe du tout et de la partie : "Le tout est nécessairement strictement plus grand que chacune de ses sous-parties strictes". C'est une notion pour laquelle je cherche à aller plus loin (dans mes travaux relativement modestes, je suis allé jusqu'aux parties de
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
Par opposition à la notion de cardinal de CANTOR c'est-à-dire la notion usuelle de cardinal , que j'appelle "cardinal potentiel" c'est-à-dire la notion de cardinal au sens de la puissance, et qui est définie pour toutes les parties de
R
n
{\displaystyle \mathbb {R} ^{n}}
principe du tout et de la partie . Donc la notion de F-quantité se veut être une notion plus fine que celle de "cardinal potentiel" c'est-à-dire que celle de cardinal (de CANTOR). Les notions de F-quantité et de "cardinal potentiel" se confondent, dans le cas des parties finies. (21-06-2024 : Pour éviter toute confusion, j'ai décidé de plutôt appeler le "cardinal quantitatif d'un ensemble" qui n'est pas, contrairement à ce que son nom laisse à penser, un cardinal (de CANTOR) d'un ensemble, la "F-quantité d'un ensemble" .)
(03-06-2021 : Rectification : La notion de F-quantité n'est pas a priori une mesure définie sur
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
(08-07-2023 : Remarque : Comme dans le cas classique de cardinal d'un ensemble, les termes "cardinal d'un ensemble" et "puissance d'un ensemble" se confondent et que l'équipotence de 2 ensembles désigne plutôt le fait que ces 2 ensembles ont même puissance, c'est-à-dire le fait que ces 2 ensembles ont même cardinal, c'est-à-dire le fait que ces 2 ensembles peuvent être mis en bijection, il est peut-être plus pertinent et plus approprié de renommer le "cardinal équipotentiel d'un ensemble" (c'est-à-dire le "cardinal d'un ensemble"), "cardinal potentiel d'un ensemble" c'est-à-dire le cardinal, au sens de la puissance, d'un ensemble, et ce, toujours, afin de le distinguer de la "F-quantité d'un ensemble" c'est-à-dire, de ce qui était anciennement nommé cardinal quantitatif ou cardinal, au sens de la quantité, d'un ensemble, même si ce n'est pas, à proprement parler, un cardinal d'un ensemble.)
(09-07-2023 : Remarque : Pour désigner le "cardinal, au sens de la puissance, d'un ensemble", je n'ai pas d'autre expression que "cardinal potentiel d'un ensemble", même si, ici, "potentiel" désigne "au sens de la puissance" et non "en puissance". Peut-être que pour l'usage que je veux en faire, il faudrait désigner le "cardinal, au sens de la puissance, d'un ensemble", "cardinal potentatif d'un ensemble" ou "cardinal potentiatif d'un ensemble", mais les termes "potentatif" et "potentiatif" sont des néologismes très rares.)
(20-09-2023 : Dans ce qui suit, j'ai remplacé l'expression "plafonnement normalisé/plafonnements normalisés" par l'expression "plafonnement normal/plafonnements normaux".)
(16-08-2024 : Dans ce qui suit, j'ai remplacé et j'ai simplifié les expressions "plafonnement borné d'une partie bornée de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
(11-11-2023 : Finalement, j’ai remplacé l'expression "axiome(s) de définition" par l'expression "hypothèse(s) de définition".)
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle PV(\mathbb {R} ^{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
σ
{\displaystyle \sigma }
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
Il est à noter qu'une partie non bornée de
R
n
{\displaystyle \mathbb {R} ^{n}}
On utilisera, essentiellement, dans la partie spéculative, une notion de limite de suites de parties de
P
V
(
R
n
)
{\displaystyle PV(\mathbb {R} ^{n})}
P
V
2
(
R
n
)
{\displaystyle PV2(\mathbb {R} ^{n})}
Comme dit ci-dessus, il y a quelques concessions à faire pour inclure le cas des sous-ensembles non bornés de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
P
V
2
(
R
″
n
)
{\displaystyle {PV2}({\mathbb {R} ''}^{n})}
nouvel espace
R
″
{\displaystyle {\mathbb {R} ''}}
, qui me semble, vu de très loin, avoir des points communs avec l'espace
∗
R
{\displaystyle *\mathbb {R} }
analyse non standard . Dans une section, j'ai essayé de définir des nombres
+
∞
f
{\displaystyle +\infty _{f}}
f
∈
F
(
R
)
{\displaystyle f\in {\cal {F}}(\mathbb {R} )}
R
″
{\displaystyle \mathbb {R} ''}
R
″
=
−
∞
F
(
R
)
⨆
R
⨆
+
∞
F
(
R
)
=
{
−
∞
f
|
f
∈
F
(
R
)
}
⨆
R
⨆
{
+
∞
f
|
f
∈
F
(
R
)
}
{\displaystyle \displaystyle {\mathbb {R} ''=-\infty _{{\cal {F}}(\mathbb {R} )}\bigsqcup \mathbb {R} \bigsqcup +\infty _{{\cal {F}}(\mathbb {R} )}=\{-\infty _{f}|f\in {\cal {F}}(\mathbb {R} )\}\bigsqcup \mathbb {R} \bigsqcup \{+\infty _{f}|f\in {\cal {F}}(\mathbb {R} )\}}}
grâce à la F-quantité [qui n'est pas un cardinal (de CANTOR)], là où le cardinal (de CANTOR) ne le peut , après avoir choisi un ensemble représentant idéal de
ℵ
0
{\displaystyle \aleph _{0}}
N
{\displaystyle \mathbb {N} }
Z
{\displaystyle \mathbb {Z} }
ℵ
1
{\displaystyle \aleph _{1}}
R
+
o
u
R
≃
P
(
N
)
{\displaystyle \mathbb {R} _{+}\,\,ou\,\,\mathbb {R} \simeq {\mathcal {P}}(\mathbb {N} )}
ℵ
2
{\displaystyle \aleph _{2}}
P
(
R
)
{\displaystyle {\mathcal {P}}(\mathbb {R} )}
Plus précisément et en particulier :
La notion de F-quantité n'est pas un cas particulier de la notion de cardinal [de CANTOR] : Elle n'a pas nécessairement de lien ou de rapport avec la notion de bijection ou avec la notion de puissance d'un ensemble ou de cardinal [de CANTOR] d'un ensemble (LA F-QUANTITÉ N'EST PAS UN CARDINAL [DE CANTOR]) .
Considérons une chaîne exhaustive de parties de
R
n
{\displaystyle \mathbb {R} ^{n}}
∅
{\displaystyle \emptyset }
R
n
{\displaystyle \mathbb {R} ^{n}}
Par convention, ici, dans cette chaîne, parmi les parties infinies de
R
n
{\displaystyle \mathbb {R} ^{n}}
F-quantité infinie d'un représentant de la puissance du dénombrable sera notée et sera égale à "
a
0
{\displaystyle a_{0}}
ℵ
0
{\displaystyle \aleph _{0}}
ℵ
0
{\displaystyle \aleph _{0}}
F-quantité infinie de
R
n
{\displaystyle \mathbb {R} ^{n}}
sera notée et sera égale à "
a
1
{\displaystyle a_{1}}
ℵ
1
{\displaystyle \aleph _{1}}
ℵ
1
{\displaystyle \aleph _{1}}
bijection , ou de puissance ou de cardinal [de CANTOR] .
"OR L'HYPOTHÈSE DU CONTINU AFFIRME QU'IL N'EXISTE AUCUN ENSEMBLE DONT LE CARDINAL [DE CANTOR] EST STRICTEMENT COMPRIS ENTRE LE CARDINAL [DE CANTOR] DE L'ENSEMBLE DES ENTIERS NATURELS ET CELUI DE L'ENSEMBLE DES NOMBRES RÉELS" . (qui est d'ailleurs indécidable dans ZFC)
Mais, par contre, il existe des ensembles dont la F-quantité [QUI N'EST PAS UN CARDINAL (DE CANTOR)] est strictement comprise entre la F-quantité de l'ensemble des entiers naturels et celle de l'ensemble des nombres réels .
Et, par convention, dans ce cas, la F-quantité de l'ensemble des entiers naturels sera notée et sera égale à "
a
0
{\displaystyle a_{0}}
ℵ
0
{\displaystyle \aleph _{0}}
ℵ
0
{\displaystyle \aleph _{0}}
F-quantité de l'ensemble des nombres réels sera notée et sera égal à "
a
1
{\displaystyle a_{1}}
ℵ
1
{\displaystyle \aleph _{1}}
ℵ
1
{\displaystyle \aleph _{1}}
(La F-quantité d'une partie non bornée de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
Michel COSTE ), mais qui y est très peu présente :
Il reste à la généraliser à des classes de parties, de plus en plus larges.
R
n
{\displaystyle \mathbb {R} ^{n}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
mais il fallait le dire avant de dire qu'une telle généralisation était impossible, au delà des parties finies .
voir supra
Mais, j'ai, maintenant, une terminologie qui rend inutiles les terminologies précédentes, je distingue, désormais, la "F-quantité" du "cardinal (de CANTOR)"
Attention : En adoptant cette terminologie, la notion de F-quantité n'est pas un cas particulier de la notion de "cardinal".
Mais sinon si on tient vraiment à attribuer le nom de "cardinal d'un ensemble" uniquement à la notion de puissance d'un ensemble qui est un ordre de grandeur de la quantité d'éléments d'un ensemble dans le cas des ensembles infinis, on peut, sans adopter la terminologie précédente, appeler, tout simplement, la notion véritable de quantité d'éléments d'un ensemble : la "F-quantité d'un ensemble".
À la place du fameux :
"Je le vois
[sous entendu : Je vois qu'il y a une bijection entre
R
{\displaystyle \mathbb {R} }
R
n
{\displaystyle \mathbb {R} ^{n}}
R
{\displaystyle \mathbb {R} }
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
[
0
;
1
]
{\displaystyle [0;1]}
mais je ne le crois pas" (de CANTOR),
je dirais plutôt :
"Je le vois
[sous-entendu : Je vois qu'il y a une bijection entre
R
{\displaystyle \mathbb {R} }
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
[
0
;
1
]
{\displaystyle [0;1]}
mais cela n'est pas suffisant
[pour caractériser la vraie notion de quantité ou de nombre d'éléments d'un ensemble infini borné ou non borné ou d'un de ses plafonnements].")
Car, par exemple, on a bien
[
−
1
,
1
]
⊊
[
−
2
,
2
]
{\displaystyle [-1,1]\subsetneq [-2,2]}
[
−
1
,
1
]
{\displaystyle [-1,1]}
[
−
2
,
2
]
{\displaystyle [-2,2]}
et on a
c
a
r
d
Q
(
[
−
2
,
2
]
∖
{
0
}
)
c
a
r
d
Q
(
[
−
1
,
1
]
∖
{
0
}
)
=
c
a
r
d
Q
(
[
−
2
,
2
]
)
−
c
a
r
d
Q
(
{
0
}
)
c
a
r
d
Q
(
[
−
1
,
1
]
)
−
c
a
r
d
Q
(
{
0
}
)
=
c
a
r
d
Q
(
[
−
2
,
2
]
)
−
1
c
a
r
d
Q
(
[
−
1
,
1
]
)
−
1
=
2
{\displaystyle \displaystyle {{\frac {{card}_{Q}([-2,2]\setminus \{0\})}{{card}_{Q}([-1,1]\setminus \{0\})}}={\frac {{card}_{Q}([-2,2])-{card}_{Q}(\{0\})}{{card}_{Q}([-1,1])-{card}_{Q}(\{0\})}}={\frac {{card}_{Q}([-2,2])-1}{{card}_{Q}([-1,1])-1}}=2}}
c
a
r
d
Q
(
[
−
1
,
1
]
)
<
c
a
r
d
Q
(
[
−
2
,
2
]
)
{\displaystyle {card}_{Q}([-1,1])<{card}_{Q}([-2,2])}
alors qu'on a
c
a
r
d
P
(
[
−
2
,
2
]
)
=
c
a
r
d
(
[
−
2
,
2
]
)
=
c
a
r
d
(
[
−
1
,
1
]
)
=
c
a
r
d
P
(
[
−
1
,
1
]
)
{\displaystyle {card}_{P}([-2,2])={card}([-2,2])={card}([-1,1])={card}_{P}([-1,1])}
c
a
r
d
Q
(
A
)
{\displaystyle {card}_{Q}(A)}
A
{\displaystyle A}
A
{\displaystyle A}
et
c
a
r
d
P
(
A
)
{\displaystyle {card}_{P}(A)}
A
{\displaystyle A}
A
{\displaystyle A}
c
a
r
d
(
A
)
{\displaystyle {card}(A)}
R
n
{\displaystyle \mathbb {R} ^{n}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
i
{\displaystyle i}
i
∈
[
[
1
,
n
]
]
{\displaystyle i\in [\![1,n]\!]}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
i
{\displaystyle i}
i
∈
[
[
1
,
n
]
]
{\displaystyle i\in [\![1,n]\!]}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
ayant une décomposition, en un nombre fini de réunions disjointes de sous-variétés compactes, convexes (connexes), simplement connexes de
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
i
{\displaystyle i}
F
i
{\displaystyle F_{i}}
i
∈
[
[
0
,
n
]
]
{\displaystyle i\in [\![0,n]\!]}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
j
{\displaystyle j}
F
i
,
j
{\displaystyle F_{i,j}}
F
i
,
j
∈
P
(
F
i
)
{\displaystyle F_{i,j}\in {\mathcal {P}}(F_{i})}
i
∈
[
[
0
,
n
]
]
{\displaystyle i\in [\![0,n]\!]}
j
∈
[
[
0
,
i
]
]
{\displaystyle j\in [\![0,i]\!]}
ou ayant une décomposition, en un nombre fini de réunions disjointes de sous-variétés ouvertes bornées, convexes (connexes), simplement connexes de
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
i
{\displaystyle i}
U
i
{\displaystyle U_{i}}
i
∈
[
[
1
,
n
]
]
{\displaystyle i\in [\![1,n]\!]}
F
0
′
∈
P
(
R
n
∖
(
⨆
i
∈
[
[
1
,
n
]
]
U
i
)
)
{\displaystyle \displaystyle {{F_{0}}'\in {\mathcal {P}}{\bigg (}\mathbb {R} ^{n}\setminus {\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}{\bigg )}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
j
{\displaystyle j}
U
i
,
j
{\displaystyle U_{i,j}}
U
i
,
j
∈
P
(
U
i
)
{\displaystyle U_{i,j}\in {\mathcal {P}}(U_{i})}
i
∈
[
[
1
,
n
]
]
{\displaystyle i\in [\![1,n]\!]}
j
∈
[
[
1
,
i
]
]
{\displaystyle j\in [\![1,i]\!]}
F
0
′
{\displaystyle {F_{0}}'}
F
0
,
0
′
∈
P
(
⨆
i
∈
[
[
1
,
n
]
]
U
i
)
{\displaystyle \displaystyle {{F_{0,0}}'\in {\mathcal {P}}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}}}
A
∈
P
(
R
n
)
{\displaystyle A\in {\mathcal {P}}(\mathbb {R} ^{n})}
telles que :
∀
i
∈
[
[
0
,
n
]
]
,
∃
F
i
{\displaystyle \forall i\in [\![0,n]\!],\exists F_{i}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
i
{\displaystyle i}
∀
i
∈
[
[
0
,
n
]
]
,
∀
j
∈
[
[
0
,
i
]
]
,
∃
F
i
,
j
{\displaystyle \forall i\in [\![0,n]\!],\,\,\forall j\in [\![0,i]\!],\,\,\exists F_{i,j}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
j
{\displaystyle j}
F
i
,
j
∈
P
(
F
i
)
{\displaystyle F_{i,j}\in {\mathcal {P}}(F_{i})}
A
=
(
⨆
i
∈
[
[
0
,
n
]
]
F
i
)
∖
(
⨆
i
∈
[
[
0
,
n
]
]
⨆
j
∈
[
[
0
,
i
]
]
F
i
,
j
)
{\displaystyle \displaystyle {A={\Big (}\bigsqcup _{i\in [\![0,n]\!]}F_{i}{\Big )}\setminus {\Big (}\bigsqcup _{i\in [\![0,n]\!]}\bigsqcup _{j\in [\![0,i]\!]}F_{i,j}{\Big )}}}
ou telles que :
∀
i
∈
[
[
1
,
n
]
]
,
∃
U
i
{\displaystyle \forall i\in [\![1,n]\!],\exists U_{i}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
i
{\displaystyle i}
∀
i
∈
[
[
1
,
n
]
]
,
∀
j
∈
[
[
1
,
i
]
]
,
∃
U
i
,
j
{\displaystyle \forall i\in [\![1,n]\!],\,\,\forall j\in [\![1,i]\!],\,\,\exists U_{i,j}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
j
{\displaystyle j}
U
i
,
j
∈
P
(
U
i
)
{\displaystyle U_{i,j}\in {\mathcal {P}}(U_{i})}
∃
F
0
′
∈
P
(
R
n
∖
(
⨆
i
∈
[
[
1
,
n
]
]
U
i
)
)
{\displaystyle \displaystyle {\exists {F_{0}}'\in {\mathcal {P}}{\bigg (}\mathbb {R} ^{n}\setminus {\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}{\bigg )}}}
∃
F
0
,
0
′
∈
P
(
⨆
i
∈
[
[
1
,
n
]
]
U
i
)
{\displaystyle \displaystyle {\exists {F_{0,0}}'\in {\mathcal {P}}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}}}
A
=
(
(
⨆
i
∈
[
[
1
,
n
]
]
U
i
)
⨆
F
0
′
)
∖
(
(
⨆
i
∈
[
[
1
,
n
]
]
⨆
j
∈
[
[
1
,
i
]
]
U
i
,
j
)
⨆
F
0
,
0
′
)
{\displaystyle \displaystyle {A={\bigg (}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}\bigsqcup {F_{0}}'{\bigg )}\setminus {\bigg (}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}\bigsqcup _{j\in [\![1,i]\!]}U_{i,j}{\Big )}\bigsqcup {F_{0,0}}'{\bigg )}}}
R
2
{\displaystyle \mathbb {R} ^{2}}
voir infra
R
n
{\displaystyle \mathbb {R} ^{n}}
Mais je pense qu'en fait, il doit être possible de comparer, entre elles, celles des parties bornées quelconques et même celles (ceux) de parties non bornées quelconques de
R
″
n
{\displaystyle {\mathbb {R} ''}^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
En effet, une fois qu'on s'est occupé de l'adhérence ou de l'intérieur d'une partie, on s'occupe ensuite de l'adhérence sans la partie ou de la partie sans l'intérieur, et on refait la même chose, avec ces dernières.
i
{\displaystyle i}
R
n
{\displaystyle \mathbb {R} ^{n}}
v
o
l
i
{\displaystyle {vol}^{i}}
(Le cas
i
=
0
{\displaystyle i=0}
https://www.fichier-pdf.fr/2021/08/07/polyintegrationmai2013/
Cf. page 13 : Chapitre 1. Les mesures/III Exemples fondamentaux d'espaces mesures/Mesures de HAUSDORFF
Cf. page 39 : Chapitre 4. La mesure de LEBESGUE et ses corollaires/II Généralisations de la mesure de LEBESGUE/II.1 Mesures de HAUSDORFF/Définition 5
Cf. page 40 : Chapitre 4. La mesure de LEBESGUE et ses corollaires/II Généralisations de la mesure de LEBESGUE/II.3 Définition alternative de la mesure de LEBESGUE/Théorème 3
Cf. page 41 : Chapitre 4. La mesure de LEBESGUE et ses corollaires/II Généralisations de la mesure de LEBESGUE/II.4 Longueur, aire, surface de parties courbées de
R
d
{\displaystyle \mathbb {R} ^{d}}
Cf. page 67 : Chapitre 7. Théorème du changement de variable/I Cas des applications linéaires
Cf. page 68 : Chapitre 7. Théorème du changement de variable/II Mesure des sous-variétés plongées
Cf. page 70 : Chapitre 7. Théorème du changement de variable/III Intégration sur les sous-variétés plongées
Cf. aussi https://homeweb.unifr.ch/manolesc/Pub/teaching/Mesure_integration.pdf
Cf. aussi https://w3.ens-rennes.fr/math/people/thibaut.deheuvels/Mesures-Hausdorff.pdf ),
i
∈
[
[
1
,
n
]
]
{\displaystyle i\in [\![1,n]\!]}
i
=
1
{\displaystyle i=1}
i
=
2
{\displaystyle i=2}
i
=
3
{\displaystyle i=3}
i
=
4
{\displaystyle i=4}
3
{\displaystyle 3}
⋯
{\displaystyle \cdots }
i
=
n
{\displaystyle i=n}
3
{\displaystyle 3}
⋯
{\displaystyle \cdots }
n
−
1
{\displaystyle n-1}
i
(
i
∈
[
[
0
,
n
]
]
)
{\displaystyle i\,\,(i\in [\![0,n]\!])}
R
n
{\displaystyle \mathbb {R} ^{n}}
v
o
l
i
~
{\displaystyle {\widetilde {vol^{i}}}}
0
{\displaystyle 0}
v
o
l
0
~
{\displaystyle {\widetilde {vol^{0}}}}
(24-06-2021 : Rectification : La notion de F-quantité n'est pas a priori une mesure définie sur
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
c
a
r
d
P
{\displaystyle {card}_{P}}
c
a
r
d
{\displaystyle {card}}
R
{\displaystyle {\mathcal {R}}}
R
2
{\displaystyle \mathbb {R} ^{2}}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
R
{\displaystyle {\mathcal {R}}}
R
2
{\displaystyle \mathbb {R} ^{2}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
2
{\displaystyle \mathbb {R} ^{2}}
R
n
{\displaystyle \mathbb {R} ^{n}}
c
a
r
d
Q
{\displaystyle {card}_{Q}}
R
{\displaystyle {\cal {R}}}
R
2
{\displaystyle \mathbb {R} ^{2}}
O
{\displaystyle O}
Nous désignons la "F-quantité, relative au repère orthonormé
R
{\displaystyle {\mathcal {R}}}
R
2
{\displaystyle \mathbb {R} ^{2}}
A
{\displaystyle A}
R
2
{\displaystyle \mathbb {R} ^{2}}
R
{\displaystyle {\mathcal {R}}}
R
2
{\displaystyle \mathbb {R} ^{2}}
A
{\displaystyle A}
R
2
{\displaystyle \mathbb {R} ^{2}}
c
a
r
d
Q
,
R
(
A
)
{\displaystyle card_{Q,{\cal {R}}}(A)}
A
{\displaystyle A}
R
2
{\displaystyle \mathbb {R} ^{2}}
c
a
r
d
P
(
A
)
{\displaystyle card_{P}(A)}
A
{\displaystyle A}
A
{\displaystyle A}
Q
F
,
R
(
A
)
{\displaystyle Q_{F,{\mathcal {R}}}(A)}
F
-
Q
R
(
A
)
{\displaystyle F{\text{-}}Q_{\mathcal {R}}(A)}
c
a
r
d
Q
,
R
(
{
O
}
×
N
n
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
3
N
)
{\displaystyle {card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {N} _{n})<{card}_{Q,{\cal {R}}}(\{O\}\times 3\mathbb {N} )}
<
c
a
r
d
Q
,
R
(
{
O
}
×
(
3
N
⨆
{
1
,
2
}
)
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
N
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
Z
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
Q
)
{\displaystyle <{card}_{Q,{\cal {R}}}{\Big (}\{O\}\times (3\mathbb {N} \bigsqcup \{1,2\}){\Big )}<{card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {N} )<{card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {Z} )<{card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {Q} )}
<
c
a
r
d
Q
,
R
(
{
O
}
×
]
−
1
,
1
[
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
[
−
1
,
1
]
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
[
−
2
,
2
]
)
{\displaystyle <card_{Q,{\cal {R}}}(\{O\}\times ]-1,1[)<{card}_{Q,{\cal {R}}}(\{O\}\times [-1,1])<{card}_{Q,{\cal {R}}}(\{O\}\times [-2,2])}
=
c
a
r
d
Q
,
R
(
{
O
}
×
(
[
−
2
,
2
]
+
1
)
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
(
(
[
−
2
,
2
]
+
1
)
⨆
{
4
}
)
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
(
R
∖
[
−
2
,
2
]
)
)
{\displaystyle ={card}_{Q,{\cal {R}}}{\Big (}\{O\}\times ([-2,2]+1){\Big )}<{card}_{Q,{\cal {R}}}{\bigg (}\{O\}\times {\Big (}([-2,2]+1)\bigsqcup \{4\}{\Big )}{\bigg )}<{card}_{Q,{\cal {R}}}{\Big (}\{O\}\times (\mathbb {R} \setminus [-2,2]){\Big )}}
<
c
a
r
d
Q
,
R
(
{
O
}
×
(
R
∖
[
−
1
,
1
]
)
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
R
∗
)
<
c
a
r
d
Q
,
R
(
{
O
}
×
R
)
{\displaystyle <{card}_{Q,{\cal {R}}}{\Big (}\{O\}\times (\mathbb {R} \setminus [-1,1]){\Big )}<{card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {R} ^{*})<{card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {R} )}
<
c
a
r
d
Q
,
R
(
[
−
1
,
1
]
×
[
−
1
,
1
]
)
<
c
a
r
d
Q
,
R
(
[
−
2
,
2
]
×
[
−
2
,
2
]
)
<
c
a
r
d
Q
,
R
(
R
2
)
{\displaystyle <{card}_{Q,{\cal {R}}}([-1,1]\times [-1,1])<{card}_{Q,{\cal {R}}}([-2,2]\times [-2,2])<{card}_{Q,{\cal {R}}}(\mathbb {R} ^{2})}
c
a
r
d
P
(
{
O
}
×
N
n
)
<
c
a
r
d
P
(
{
O
}
×
3
N
)
{\displaystyle {card}_{P}(\{O\}\times \mathbb {N} _{n})<{card}_{P}(\{O\}\times 3\mathbb {N} )}
=
c
a
r
d
P
(
{
O
}
×
(
3
N
⨆
{
1
,
2
}
)
)
=
c
a
r
d
P
(
{
O
}
×
N
)
=
c
a
r
d
P
(
{
O
}
×
Z
)
=
c
a
r
d
P
(
{
O
}
×
Q
)
{\displaystyle ={card}_{P}{\Big (}\{O\}\times (3\mathbb {N} \bigsqcup \{1,2\}){\Big )}={card}_{P}(\{O\}\times \mathbb {N} )={card}_{P}(\{O\}\times \mathbb {Z} )={card}_{P}(\{O\}\times \mathbb {Q} )}
<
c
a
r
d
P
(
{
O
}
×
]
−
1
,
1
[
)
=
c
a
r
d
P
(
{
O
}
×
[
−
1
,
1
]
)
=
c
a
r
d
P
(
{
O
}
×
[
−
2
,
2
]
)
{\displaystyle <{card}_{P}(\{O\}\times ]-1,1[)={card}_{P}(\{O\}\times [-1,1])={card}_{P}(\{O\}\times [-2,2])}
=
c
a
r
d
P
(
{
O
}
×
(
[
−
2
,
2
]
+
1
)
)
=
c
a
r
d
P
(
{
O
}
×
(
(
[
−
2
,
2
]
+
1
)
⨆
{
4
}
)
)
=
c
a
r
d
P
(
{
O
}
×
(
R
∖
[
−
2
,
2
]
)
)
{\displaystyle ={card}_{P}{\Big (}\{O\}\times ([-2,2]+1){\Big )}={card}_{P}{\bigg (}\{O\}\times {\Big (}([-2,2]+1)\bigsqcup \{4\}{\Big )}{\bigg )}={card}_{P}{\Big (}\{O\}\times (\mathbb {R} \setminus [-2,2]){\Big )}}
=
c
a
r
d
P
(
{
O
}
×
(
R
∖
[
−
1
,
1
]
)
)
=
c
a
r
d
P
(
{
O
}
×
R
∗
)
=
c
a
r
d
P
(
{
O
}
×
R
)
{\displaystyle ={card}_{P}{\Big (}\{O\}\times (\mathbb {R} \setminus [-1,1]){\Big )}={card}_{P}(\{O\}\times \mathbb {R} ^{*})={card}_{P}(\{O\}\times \mathbb {R} )}
=
c
a
r
d
P
(
[
−
1
,
1
]
×
[
−
1
,
1
]
)
=
c
a
r
d
P
(
[
−
2
,
2
]
×
[
−
2
,
2
]
)
=
c
a
r
d
P
(
R
2
)
{\displaystyle ={card}_{P}([-1,1]\times [-1,1])={card}_{P}([-2,2]\times [-2,2])={card}_{P}(\mathbb {R} ^{2})}
2) Dans une bouteille de
2
L
{\displaystyle 2L}
1
L
{\displaystyle 1L}
Je viens de donner la raison d'être et l'utilité de la notion de F-quantité.
On ne fait pas toujours des mathématiques, en vue d'applications pratiques ou concrètes.
Pourtant à qui lui veut des applications :
La notion de quantité de matière discrète ou de matière continue, parle d'elle-même.
Supposons qu'un univers soit fait d'un mélange de matière continue et de matière discrète :
La F-quantité mesure la quantité de matière continue et de matière discrète.
La notion de matière continue n'existe certes pas dans notre univers, mais on peut la concevoir mathématiquement et c'est une bonne approximation de la matière discrète, à l'échelle macroscopique, en physique.
La notion de (F-)quantité est plus fine que celle de puissance qui donne, seulement, un ordre de grandeur de la première.
[Rectification : En fait, tout dépend des "plafonnements" de chacun des 2 disques durs cubiques compacts et plus généralement des "plafonnements" des parties infinies bornées que l'on s'est fixé et, plus particulièrement, des densités (quantitatives) uniformes ou pas, que l'on s'est fixé, des "matières continues et/ou discrètes" qui les composent et qui sont composées chacune au moins d'une infinité de points de "matière continue" et/ou de "matière discrète"
(Tout point étant de dimension nulle, les interprétations concernant les densités quantitatives des parties infinies bornées sont multiples voire infinies et donc aussi concernant leurs F-quantités
[relatives à un repère orthonormé de
R
n
{\displaystyle \mathbb {R} ^{n}}
relativement aux plafonnements et selon les plafonnements que l'on s'est fixé).
Remarque : Cela marche aussi avec les "plafonnements" des parties (infinies) non bornées.
Il existe, néanmoins, pour chaque partie bornée, un ou des plafonnement(s), et pour chaque partie non bornée, un ou des plafonnement(s), dits normaux.]
Restera à généraliser cette notion aux parties de
P
(
R
n
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})}
P
(
P
(
R
n
)
)
{\displaystyle {\mathcal {P}}{\Big (}{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
i
(
0
≤
i
≤
n
)
{\displaystyle i\,\,(0\leq i\leq n)}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
{\displaystyle \mathbb {R} }
R
n
{\displaystyle \mathbb {R} ^{n}}
Comment généraliser ces notions ou trouver des notions affaiblies qui marchent, aussi, dans d'autres espaces, par exemple sur des espaces qui dépendent de
R
{\displaystyle \mathbb {R} }
Le PDF : "La saga du "cardinal"" (version 4) de Michel COSTE guide le lecteur en expliquant intuitivement les notions et les idées qu'il présente ainsi que tout le cheminement qui a permis d'y aboutir à travers des exemples.
Le but de mes travaux n'est pas, mise à part l'introduction, de reproduire et d'inclure ou d'incorporer tout le travail d'explication, d'explicitation, d'illustration, de vulgarisation et de pédagogie effectué par Michel COSTE ainsi que toute la prise par la main du lecteur par ce dernier, mais d'enchaîner rigoureusement les définitions, propositions, résultats et exemples comme cela est le cas dans de nombreux livres de mathématiques, même si ceux-ci sont censés donner une certaine idée et une certaine intuition des objets manipulés.
Le PDF informel de vulgarisation de Michel COSTE répond aux attentes {des amateurs|de l'amateur}, mais il ne répond pas à toutes les attentes {des mathématiciens|du mathématicien}.
Il faut peut-être que je travaille encore l'énoncé d'un des théorèmes de mes travaux et que je le distingue bien de sa démonstration.
Depuis quelques temps, j'ai fait un travail censé éclaircir et désambiguïser les hypothèses de définition de la F-quantité en précisant rigoureusement pour chacune d'entre elles, son domaine {d'application|de validité} respectif, certains domaines étant plus généraux que d'autres, mais au final on a toutes les hypothèses de définition dont on a besoin sur le domaine
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
Mes travaux n'ont pas par exemple pour but comme Michel COSTE l'a fait à partir du théorème de STEINER-MINKOWSKI, d'expliquer géométriquement la nature des coefficients qui interviennent dans la formule de la F-quantité sur
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
Mais, peut-être que je dois encore intervenir dans son contenu et dans sa forme, pour la mettre dans une forme qui satisfasse les intervenants Des-mathematiques.net, en m'inspirant du PDF de Michel COSTE.
Mais, je n'aurais pas pu faire, de moi-même, la vulgarisation qu'a faite Michel COSTE dans son PDF, car je ne disposais pas de tous les éléments et de toutes les connaissances pour le faire, et, pour les mêmes raisons, j'ai des limites à pouvoir faire mieux que lui et à compléter son travail, concernant la partie connue et établie.
Il est vrai que mes travaux sur la F-quantité sont beaucoup plus secs que le PDF de Michel COSTE, "La saga du "cardinal"" : Je ne dis pas que tout ce qu'a dit dedans Michel COSTE est inutile et n'aide pas à la compréhension, mais si on veut démontrer ou utiliser de manière opérationnelle les résultats qui y sont mentionnés, on n'a pas besoin de tous les commentaires qu'il y a faits.
Par ailleurs, lorsque j'ai posté mes travaux sur la F-quantité et autres sur Les-mathematiques.net (Je viens de faire supprimer un certain nombre de pages, il reste encore la version 3 du PDF de Michel COSTE), je me suis quasiment comporté comme s'il s'agissait d'une page de brouillon, d'où le déchaînement et la déferlante de critiques, d'interprétations, de malentendus et de conclusions parfois et même souvent faux, erronés, hâtifs, malvenus ou infondés qu'ils ont pu susciter y compris sur ma propre personne et mes propres compétences et capacités en mathématiques, même si par ailleurs une partie était parfaitement justifiée.
D'une manière générale, lorsque je me suis lancé dans des travaux peu académiques et non balisés, j'ai vraiment eu de bonnes intuitions.
Mais lorsqu'il s'agit de les exprimer, de les préciser et de les affiner, je suis susceptible d'écrire plein d'âneries et de conneries, pendant une longue période voire une très longue période, même lorsque je dispose des connaissances pour les éviter, conneries qui se résorbent et se résorberont peu à peu, jusqu'à finir et/ou jusqu'à peut-être finir par faire aboutir mes intuitions initiales.
Cette façon de faire et de procéder ne passe pas inaperçue et ne passe malheureusement pas et visiblement pas sur Les-mathematiques.net et sur Maths-Forum, et y faisait désordre.
Certaines de mes discussions hors F-quantité et certains délires et divagations auraient dû être évités et auraient dû rester de l'ordre du brouillon personnel.
La situation de mes travaux sur Les-mathematiques.net est, de toute façon, devenue pourrie et irrécupérable, quels que soient les éventuels avancements ou progrès que j'aurais faits ou que je ferai à l'avenir.
Si l'ensemble
+
∞
F
(
R
)
{\displaystyle +\infty _{{\mathcal {F}}(\mathbb {R} )}}
J'ai de bonnes raisons de croire que la sous-section restante de la partie spéculative est valable et bonne dans le fond, et qu'il y a juste à intervenir encore dans son contenu et dans sa forme, pourvu que la définition de limite d'une famille de parties de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
[26-09-2023 : La notion de plafonnement d'une partie de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
N'oubliez pas de consulter : https://www.philo-et-societe-2-0.com/
REMARQUE : On pourra d'abord lire les PDF de Michel COSTE, qui sont des articles informels de vulgarisation, beaucoup moins ambitieux :
Michel COSTE sur Les-mathematiques.net à propos de mes travaux en 2007 :
i
(
0
≤
i
≤
n
)
{\displaystyle i(0\leq i\leq n)}
R
n
{\displaystyle \mathbb {R} ^{n}}
i
=
n
{\displaystyle i=n}
"Partie déjà établie et connue : F-quantité définie sur
P
V
(
R
n
)
{\displaystyle {PV}({\mathbb {R} }^{n})}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
R
N
{\displaystyle \mathbb {R} ^{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
.
Cf. Référence:Géométrie (BERGER)
Michel COSTE , il provient de Jean DIEUDONNÉ :
Voici des liens Wikipedia :
Voici des liens intéressants en français :
Voici un lien intéressant en anglais (du moins le début, en ce qui me concerne) :
La notion de F-quantité sur
R
n
{\displaystyle \mathbb {R} ^{n}}
Voici des liens dont il faut vraiment éviter de consulter les pages concernées :
En mai 2021, sous un compte "MPF" créé à cet effet, j'avais demandé à Lostounet, l'un des administrateurs du forum Maths-Forum, de supprimer, en lui listant les liens url, les discussions que j'avais initiées et créées, il y a 4-5 ans, relatives à la F-quantité, car elles font de l'ombre à la version actualisée de mes travaux sur (la) Wikiversité.
Or celui-ci n'a pas exécuté ma demande et a préféré, à la place et sans que je lui ai demandé, supprimer mon compte "Matheux philosophe" avec tous ses messages et m'a banni après, seulement, 3 messages, sous mon compte "MPF".
NB : J'avais déjà été banni sous mon pseudo "Matheux philosophe" à cause de ces discussions et du fait que j'avais signalé que Les-mathematiques.net m'avaient déjà banni pour des discussions antérieures sur le même thème.
Cf. aussi Utilisateur:Guillaume FOUCART/Passages que l'on peut omettre/A propos de ma demande de suppression de discussions sur le forum Maths-Forum
ou (version complète avec mes messages)
ou (version complète avec mes messages)
sauf concernant 2 messages : 1 et 2
Voici un message d'une discussion sur Les-mathematiques.net, où je réponds à certaines critiques :
Voici un message sur le forum Futura-Sciences où l'intervenant Médiat a eu tort en disant qu'en qualifiant ma notion de "cardinal quantitatif", cela sous-entendait que je qualifiais la notion de cardinal classique (ou de CANTOR) de "cardinal qualitatif". Or il n'en est rien, puisque si j'ai qualifié ma notion de "cardinal quantitatif", c'est en effet pour l'opposer au cardinal classique (ou de CANTOR), mais en qualifiant ce dernier de "cardinal potentiel", tout en sachant que le cardinal quantitatif est la {vraie|véritable} notion de quantité d'éléments d'un ensemble alors que le cardinal classique (ou de CANTOR) n'est en fait qu'un ordre de grandeur de la quantité d'éléments concernant les ensembles infinis :
Utilisateur:Guillaume FOUCART/Passages que l'on peut omettre/Commentaires, impressions voire spéculations autour des amateurs, des shtameurs, de moi-même, des intervenants et des grands intervenants sur les forums de mathématiques
NB : Michel COSTE, qui tient à sa réputation, est uniquement responsable de ses propres propos dans les PDF dont il est l'auteur c'est-à-dire, ici, dans les documents intitulés "La saga du "cardinal"" (versions 1-2-3-4), qui sont des articles informels de vulgarisation.
Avant d'envisager la formule de la F-quantité concernant les parties bornées de
R
″
n
{\displaystyle \mathbb {R} ''^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
NB : le principal et le plus dur reste encore à faire.
On pourra peut-être ensuite l'étendre à des classes de parties de
R
″
n
{\displaystyle {\mathbb {R} ''}^{n}}
Je sais que si des suites de polytopes de
R
n
{\displaystyle \mathbb {R} ^{n}}
n
{\displaystyle n}
R
n
{\displaystyle \mathbb {R} ^{n}}
n
{\displaystyle n}
n
{\displaystyle n}
(Cf. articles informels de vulgarisation de Michel COSTE que j'ai donnés (voir supra
Le début des versions 1, 2 et 3, contient un passage que l'auteur a préféré supprimer dans la version 4, mais ce passage est fondamental pour moi, et est caractéristique et constitutif de la {vraie|véritable} notion de quantité d'éléments d'un ensemble, et qui dit que cette notion, appliquée à un ensemble, ne néglige aucun point, et que la F-quantité de tout singleton de
R
n
{\displaystyle \mathbb {R} ^{n}}
1
{\displaystyle 1}
La documentation disponible tourne autour de la géométrie convexe et de la formule de STEINER-MINKOWSKI qui est fausse dans le cas des parties non convexes, mais cela est insuffisant voire inutile, si on veut aller au-delà des parties convexes.
Je sais que tout polyèdre non convexe est décomposable en polyèdres convexes. Il y a donc peut-être là une possibilité d'étendre la notion de F-quantité en supprimant la contrainte de convexité de ma définition des PV.
"Toute partie non convexe, connexe, de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
donc toute partie non convexe, de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
donc toute partie de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
3
{\displaystyle 3}
R
3
{\displaystyle \mathbb {R} ^{3}}
2
{\displaystyle 2}
La plupart des surfaces de classe
C
1
{\displaystyle {\mathcal {C}}^{1}}
R
3
{\displaystyle \mathbb {R} ^{3}}
2
{\displaystyle 2}
Certaines surfaces de
R
3
{\displaystyle \mathbb {R} ^{3}}
2
{\displaystyle 2}
Certaines surfaces de classe
C
1
{\displaystyle {\mathcal {C}}^{1}}
0
{\displaystyle 0}
Michel COSTE et Denis FELDMANN disent pour l'un qu'ils ne peuvent raisonnablement pas aller au-delà des PV, et pour l'autre au-delà des parties bornées de
R
n
{\displaystyle \mathbb {R} ^{n}}
Michel COSTE a vu et a fait le lien et le rapprochement entre la F-quantité et la formule de STEINER-MINKOWSKI, mais tous les travaux qui tournent autour de cette formule concernent principalement, le théorème de HADWIGER, les inégalités isopérimétriques, l'inégalité de BRUNN-MINKOWSKI et la formule de PICK et ignorent complètement, mais peut-être pas, totalement, pour le 1er, la notion que je cherche à étendre.
Par ailleurs, j'ai introduit des notions qui sont peut-être inutiles pour étendre la F-quantité aux "seules" parties de
R
n
{\displaystyle \mathbb {R} ^{n}}
De plus, il se peut qu'elles aient été déjà inventées par d'autres personnes, avant moi, mais dans tous les cas, on devrait, normalement, leur trouver une utilité.
Pour le moment, je sais comparer les F-quantités, au moins, des PV de
R
n
{\displaystyle \mathbb {R} ^{n}}
i
(
0
≤
i
≤
n
)
{\displaystyle i\,\,(0\leq i\leq n)}
R
″
n
{\displaystyle {\mathbb {R} ''}^{n}}
i
(
0
≤
i
≤
n
)
{\displaystyle i\,\,(0\leq i\leq n)}
Partie déjà établie et connue : F-quantité définie sur
P
V
(
R
n
)
{\displaystyle {PV}({\mathbb {R} }^{n})}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[en fait, à un changement de notion de limite de famille de parties de
R
n
{\displaystyle \mathbb {R} ^{n}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
[ modifier | modifier le wikicode ] En fait, puisque la "F-quantité, relative au repère orthonormé
R
{\displaystyle {\mathcal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
{\displaystyle {\mathcal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
Q
F
,
R
(
A
)
{\displaystyle Q_{F,{\mathcal {R}}}(A)}
F
-
Q
R
(
A
)
{\displaystyle F{\text{-}}Q_{\mathcal {R}}(A)}
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q,{\mathcal {R}}}(A)}
J'ai admis les propositions et les théorèmes pour lesquels Michel COSTE n'a pas fourni de démonstration ou n'a pas donné de référence [Ce sont, sans doute, les démonstrations les plus difficiles qui permettraient, au lecteur, d'attacher plus d'importance et de crédit, et de donner, d'avantage, corps à cette théorie].
P
o
l
y
t
o
p
e
s
(
R
N
)
{\displaystyle {{\cal {P}}olytopes}(\mathbb {R} ^{N})}
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
P
o
l
y
t
o
p
e
s
(
R
N
)
{\displaystyle {{\cal {P}}olytopes}(\mathbb {R} ^{N})}
=
d
e
´
f
{
P
N
∈
P
(
R
N
)
|
P
N
p
o
l
y
t
o
p
e
d
e
R
N
(
c
-
a
`
-
d
p
o
l
y
e
`
d
r
e
c
o
m
p
a
c
t
,
c
o
n
v
e
x
e
[
c
o
n
n
e
x
e
]
d
e
R
N
)
}
⋃
{
∅
}
{\displaystyle \displaystyle {{\underset {d{\acute {e}}f}{=}}{\Big \{}{P^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{P^{N}}\,\,polytope\,\,de\,\,\mathbb {R} ^{N}\,\,(c{\mbox{-}}{\grave {a}}{\mbox{-}}d\,\,poly{\grave {e}}dre\,\,compact,\,\,convexe\,\,[connexe]\,\,de\,\,\mathbb {R} ^{N}){\Big \}}\bigcup \{\emptyset \}}}
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
=
d
e
´
f
{
A
N
∈
P
(
R
N
)
|
A
N
s
o
u
s
-
v
a
r
i
e
´
t
e
´
c
o
m
p
a
c
t
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
d
e
R
N
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
o
u
s
a
n
s
b
o
r
d
}
{\displaystyle {\underset {d{\acute {e}}f}{=}}{\Big \{}{A^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{A^{N}}\,\,sous{\text{-}}vari{\acute {e}}t{\acute {e}}\,\,compacte,\,\,convexe\,\,(connexe)\,\,de\,\,\mathbb {R} ^{N},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,ou\,\,sans\,\,bord{\Big \}}}
Fin du théorème
Remarque : La topologie choisie, ici, est la topologie de HAUSDORFF.
P
(
R
n
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})}
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
Soit
R
{\displaystyle {\cal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
O
{\displaystyle O}
On pose :
P
b
o
r
n
e
´
e
s
(
R
n
)
=
d
e
´
f
{
A
∈
P
(
R
n
)
|
A
b
o
r
n
e
´
e
d
a
n
s
R
n
}
{\displaystyle {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n}){\underset {d{\acute {e}}f}{=}}\{A\in {\mathcal {P}}({\mathbb {R} }^{n})\,\,|\,\,A\,\,born{\acute {e}}e\,\,dans\,\,\mathbb {R} ^{n}\}}
R
{\displaystyle {\mathcal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
la restriction à l'ensemble
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
et la restriction à l'ensemble
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
c
a
r
d
Q
,
R
:
P
(
R
n
)
⟶
(
F
″
,
+
,
×
,
≤
)
{\displaystyle {card}_{Q,{\cal {R}}}:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\longrightarrow \,\,(F'',+,\times ,\leq )}
c
a
r
d
Q
,
R
|
P
b
o
r
n
e
´
e
s
(
R
n
)
:
P
b
o
r
n
e
´
e
s
(
R
n
)
⟶
(
F
′
,
+
,
×
,
≤
)
{\displaystyle {{card}_{Q,{\cal {R}}}}_{|{\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}\,\,:\,\,{\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})\,\,\longrightarrow \,\,(F',+,\times ,\leq )}
c
a
r
d
Q
,
R
|
P
V
(
R
n
)
:
P
V
(
R
n
)
⟶
(
F
,
+
,
×
,
≤
)
{\displaystyle {{card}_{Q,{\cal {R}}}}_{|{PV}(\mathbb {R} ^{n})}\,\,:\,\,{PV}({\mathbb {R} }^{n})\,\,\longrightarrow \,\,(F,+,\times ,\leq )}
c
a
r
d
Q
,
R
|
P
o
l
y
t
o
p
e
s
(
R
n
)
:
P
o
l
y
t
o
p
e
s
(
R
n
)
⟶
(
F
,
+
,
×
,
≤
)
{\displaystyle {{card}_{Q,{\cal {R}}}}_{|{{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}\,\,:\,\,{{\mathcal {P}}olytopes}(\mathbb {R} ^{n})\,\,\longrightarrow \,\,(F,+,\times ,\leq )}
(
F
″
,
+
,
×
,
≤
)
{\displaystyle (F'',+,\times ,\leq )}
où
(
F
′
,
+
,
×
,
≤
)
{\displaystyle (F',+,\times ,\leq )}
où
(
F
,
+
,
×
,
≤
)
{\displaystyle (F,+,\times ,\leq )}
F
=
R
n
[
c
a
r
d
Q
,
R
(
I
)
]
{\displaystyle F=\mathbb {R} _{n}[{card}_{Q,{\mathcal {R}}}(I)]}
I
{\displaystyle I}
R
{\displaystyle \mathbb {R} }
I
=
[
0
;
1
[
{\displaystyle I=[0;1[}
et où
F
⊂
F
′
⊂
F
″
{\displaystyle F\subset F'\subset F''}
F
′
=
?
,
F
″
=
?
{\displaystyle F'=?,\,\,F''=?}
c
a
r
d
Q
,
R
:
P
(
R
n
)
⟶
N
⨆
+
∞
{\displaystyle \displaystyle {{{card}_{Q,{\cal {R}}}}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\longrightarrow \,\,\mathbb {N} \bigsqcup +\infty }}
c
a
r
d
Q
,
R
|
P
b
o
r
n
e
´
e
s
(
R
n
)
:
P
b
o
r
n
e
´
e
s
(
R
n
)
⟶
N
⨆
+
∞
{\displaystyle \displaystyle {{{card}_{Q,{\cal {R}}}}_{|{\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}\,\,:\,\,{\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})\,\,\longrightarrow \,\,\mathbb {N} \bigsqcup +\infty }}
c
a
r
d
Q
,
R
|
P
V
(
R
n
)
:
P
V
(
R
n
)
⟶
N
⨆
+
∞
{\displaystyle \displaystyle {{{card}_{Q,{\cal {R}}}}_{|{PV}(\mathbb {R} ^{n})}\,\,:\,\,{PV}({\mathbb {R} }^{n})\,\,\longrightarrow \,\,\mathbb {N} \bigsqcup +\infty }}
et
c
a
r
d
Q
,
R
|
P
o
l
y
t
o
p
e
s
(
R
n
)
:
P
o
l
y
t
o
p
e
s
(
R
n
)
⟶
N
⨆
+
∞
{\displaystyle \displaystyle {{{card}_{Q,{\cal {R}}}}_{|{{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}\,\,:\,\,{{\mathcal {P}}olytopes}(\mathbb {R} ^{n})\,\,\longrightarrow \,\,\mathbb {N} \bigsqcup +\infty }}
de manière non classique , on considère : "
+
∞
{\displaystyle +\infty }
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle +\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
qui doivent, normalement, vérifier les conditions suivantes (Règles et opérations générales sur la F-quantité) :
∀
R
,
R
′
{\displaystyle \forall {\mathcal {R}},{\mathcal {R}}'}
R
n
,
c
a
r
d
Q
,
R
|
P
b
o
r
n
e
´
e
s
(
R
n
)
=
c
a
r
d
Q
,
R
′
|
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle \mathbb {R} ^{n},\,\,{{card}_{Q,{\mathcal {R}}}}_{|{\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}={{card}_{Q,{\mathcal {R}}'}}_{|{\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}}
On pose donc :
∀
R
{\displaystyle \forall {\mathcal {R}}}
R
n
,
c
a
r
d
Q
,
R
|
P
b
o
r
n
e
´
e
s
(
R
n
)
=
c
a
r
d
Q
~
~
{\displaystyle \mathbb {R} ^{n},\,\,{{card}_{Q,{\mathcal {R}}}}_{|{\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}={\widetilde {\widetilde {{card}_{Q}}}}}
c
a
r
d
Q
,
R
|
P
V
(
R
n
)
=
c
a
r
d
Q
~
~
|
P
V
(
R
n
)
=
c
a
r
d
Q
~
{\displaystyle \displaystyle {{{card}_{Q,{\mathcal {R}}}}_{|PV(\mathbb {R} ^{n})}={\widetilde {\widetilde {{card}_{Q}}}}_{|{PV}(\mathbb {R} ^{n})}={\widetilde {{card}_{Q}}}}}
[a)
∀
A
∈
P
(
R
n
)
{\displaystyle \forall A\in {\mathcal {P}}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
(
A
)
≥
0
{\displaystyle card_{Q,{\cal {R}}}(A)\geq 0}
b)
c
a
r
d
Q
,
R
(
∅
)
=
0
{\displaystyle card_{Q,{\cal {R}}}(\emptyset )=0}
c)
∀
x
∈
R
n
,
c
a
r
d
Q
,
R
(
{
x
}
)
=
1
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,{card}_{Q,{\cal {R}}}(\{x\})=1}
∀
A
,
B
∈
P
(
R
n
)
{\displaystyle \forall A,B\in {\mathcal {P}}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
(
A
⋃
B
)
=
c
a
r
d
Q
,
R
(
A
)
+
c
a
r
d
Q
,
R
(
B
)
−
c
a
r
d
Q
,
R
(
A
⋂
B
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(A\bigcup B)={card}_{Q,{\cal {R}}}(A)+{card}_{Q,{\cal {R}}}(B)-{card}_{Q,{\cal {R}}}(A\bigcap B)}}
donc, en particulier,
∀
A
,
B
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle \forall A,B\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
(
A
⋃
B
)
=
c
a
r
d
Q
,
R
(
A
)
+
c
a
r
d
Q
,
R
(
B
)
−
c
a
r
d
Q
,
R
(
A
⋂
B
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(A\bigcup B)={card}_{Q,{\cal {R}}}(A)+{card}_{Q,{\cal {R}}}(B)-{card}_{Q,{\cal {R}}}(A\bigcap B)}}
∀
(
x
m
)
m
∈
N
⊂
R
,
c
o
n
v
e
r
g
e
n
t
e
d
a
n
s
R
,
lim
m
→
+
∞
c
a
r
d
Q
,
R
(
[
0
,
x
m
]
)
=
c
a
r
d
Q
,
R
(
[
0
,
lim
m
→
+
∞
x
m
]
)
{\displaystyle \displaystyle {\forall {(x_{m})}_{m\in \mathbb {N} }\subset {\mathbb {R} },\,\,convergente\,\,dans\,\,\mathbb {R} ,\,\,\lim _{m\rightarrow +\infty }{card}_{Q,{\cal {R}}}([0,x_{m}])={card}_{Q,{\cal {R}}}([0,\lim _{m\rightarrow +\infty }x_{m}])}}
∀
i
∈
N
n
∗
,
R
i
{\displaystyle \forall i\in \mathbb {N} _{n}^{*},\,\,{\cal {R}}_{i}}
R
i
{\displaystyle {\mathbb {R} }^{i}}
O
i
(
0
)
j
∈
N
i
∗
{\displaystyle O_{i}{(0)}_{j\in \mathbb {N} _{i}^{*}}}
∀
k
∈
N
n
−
1
∗
{\displaystyle \forall k\in \mathbb {N} _{n-1}^{*}}
∀
A
∈
P
b
o
r
n
e
´
e
s
(
R
n
−
k
)
{\displaystyle \displaystyle {\forall A\in {\mathcal {P}}_{born{\acute {e}}es}({\mathbb {R} }^{n-k})}}
∀
B
∈
P
b
o
r
n
e
´
e
s
(
R
k
)
{\displaystyle \displaystyle {\forall B\in {\mathcal {P}}_{born{\acute {e}}es}({\mathbb {R} }^{k})}}
c
a
r
d
Q
,
R
(
A
×
B
)
=
c
a
r
d
Q
,
R
n
(
A
×
B
)
=
c
a
r
d
Q
,
R
n
−
k
(
A
)
c
a
r
d
Q
,
R
k
(
B
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(A\times B)={card}_{Q,{\cal {R}}_{n}}(A\times B)={card}_{Q,{\cal {R}}_{n-k}}(A)\,\,{card}_{Q,{\cal {R}}_{k}}(B)}}
∀
k
∈
N
n
−
1
∗
{\displaystyle \forall k\in \mathbb {N} _{n-1}^{*}}
∀
A
∈
P
o
l
y
t
o
p
e
s
(
R
n
−
k
)
{\displaystyle \displaystyle {\forall A\in {{\mathcal {P}}olytopes}({\mathbb {R} }^{n-k})}}
∀
B
∈
P
o
l
y
t
o
p
e
s
(
R
k
)
{\displaystyle \displaystyle {\forall B\in {{\mathcal {P}}olytopes}({\mathbb {R} }^{k})}}
c
a
r
d
Q
,
R
(
A
×
B
)
=
c
a
r
d
Q
,
R
n
(
A
×
B
)
=
c
a
r
d
Q
,
R
n
−
k
(
A
)
c
a
r
d
Q
,
R
k
(
B
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(A\times B)={card}_{Q,{\cal {R}}_{n}}(A\times B)={card}_{Q,{\cal {R}}_{n-k}}(A)\,\,{card}_{Q,{\cal {R}}_{k}}(B)}}
@Attention, concernant les parties non bornées, les formules ci-dessus, n'ont pas, nécessairement, sens, si on suppose qu'il y a un plafonnement sphérique, autour de l'origine du repère orthonormé direct
R
{\displaystyle {\cal {R}}}
5)
∀
I
intervalle de
R
n
{\displaystyle \forall I\,\,{\mbox{intervalle de}}\,\,\mathbb {R} ^{n}}
I
∈
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle I\in {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
∀
i
s
isométrie de
R
n
{\displaystyle \forall {is}\,\,{\mbox{isométrie de}}\,\,\mathbb {R} ^{n}}
c
a
r
d
Q
,
R
(
i
s
(
I
)
)
=
c
a
r
d
Q
,
R
(
I
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{is}(I){\Big )}={card}_{Q,{\cal {R}}}(I)}
donc, en particulier,
∀
I
intervalle de
R
n
{\displaystyle \forall I\,\,{\mbox{intervalle de}}\,\,\mathbb {R} ^{n}}
I
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle I\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
i
s
isométrie de
R
n
{\displaystyle \forall {is}\,\,{\mbox{isométrie de}}\,\,\mathbb {R} ^{n}}
c
a
r
d
Q
,
R
(
i
s
(
I
)
)
=
c
a
r
d
Q
,
R
(
I
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{is}(I){\Big )}={card}_{Q,{\cal {R}}}(I)}
∀
I
intervalle de
R
n
{\displaystyle \forall I\,\,{\mbox{intervalle de}}\,\,\mathbb {R} ^{n}}
I
∈
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle I\in {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
∀
x
∈
R
n
,
c
a
r
d
Q
,
R
(
t
x
(
I
)
)
=
c
a
r
d
Q
,
R
(
I
)
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,{card}_{Q,{\cal {R}}}{\Big (}t_{x}(I){\Big )}={card}_{Q,{\cal {R}}}(I)}
où
∀
x
∈
R
n
,
t
x
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
A
+
x
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,t_{x}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,A+x}
x
{\displaystyle x}
R
n
{\displaystyle \mathbb {R} ^{n}}
donc, en particulier,
∀
I
intervalle de
R
n
{\displaystyle \forall I\,\,{\mbox{intervalle de}}\,\,\mathbb {R} ^{n}}
I
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle I\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
x
∈
R
n
,
c
a
r
d
Q
,
R
(
t
x
(
I
)
)
=
c
a
r
d
Q
,
R
(
I
)
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,{card}_{Q,{\cal {R}}}{\Big (}t_{x}(I){\Big )}={card}_{Q,{\cal {R}}}(I)}
où
∀
x
∈
R
n
,
t
x
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
A
+
x
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,t_{x}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,A+x}
x
{\displaystyle x}
R
n
{\displaystyle \mathbb {R} ^{n}}
∀
I
intervalle de
R
n
{\displaystyle \forall I\,\,{\mbox{intervalle de}}\,\,\mathbb {R} ^{n}}
I
∈
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle I\in {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
∀
M
∈
R
n
{\displaystyle \displaystyle {\forall M\in \mathbb {R} ^{n}}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
c
a
r
d
Q
,
R
(
r
o
t
(
M
,
θ
n
)
(
I
)
)
=
c
a
r
d
Q
,
R
(
I
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{rot}_{(M,\theta _{n})}(I){\Big )}={card}_{Q,{\cal {R}}}(I)}
où
∀
M
∈
R
n
{\displaystyle \displaystyle {\forall M\in \mathbb {R} ^{n}}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
r
o
t
(
M
,
θ
n
)
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
r
o
t
(
M
,
θ
n
)
(
A
)
{\displaystyle {rot}_{(M,\theta _{n})}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,{rot}_{(M,\theta _{n})}(A)}
M
{\displaystyle M}
θ
n
{\displaystyle \theta _{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
donc, en particulier,
∀
I
intervalle de
R
n
{\displaystyle \forall I\,\,{\mbox{intervalle de}}\,\,\mathbb {R} ^{n}}
I
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle I\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
M
∈
R
n
{\displaystyle \displaystyle {\forall M\in \mathbb {R} ^{n}}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
c
a
r
d
Q
,
R
(
r
o
t
(
M
,
θ
n
)
(
I
)
)
=
c
a
r
d
Q
,
R
(
I
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{rot}_{(M,\theta _{n})}(I){\Big )}={card}_{Q,{\cal {R}}}(I)}
où
∀
M
∈
R
n
{\displaystyle \displaystyle {\forall M\in \mathbb {R} ^{n}}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
r
o
t
(
M
,
θ
n
)
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
r
o
t
(
M
,
θ
n
)
(
A
)
{\displaystyle {rot}_{(M,\theta _{n})}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,{rot}_{(M,\theta _{n})}(A)}
M
{\displaystyle M}
θ
n
{\displaystyle \theta _{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
a)
∀
A
∈
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle \forall A\in {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
∀
i
s
isométrie de
R
n
{\displaystyle \forall {is}\,\,{\mbox{isométrie de}}\,\,\mathbb {R} ^{n}}
c
a
r
d
Q
,
R
(
i
s
(
A
)
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{is}(A){\Big )}={card}_{Q,{\cal {R}}}(A)}
donc, en ppaticulier,
∀
A
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle \forall A\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
i
s
isométrie de
R
n
{\displaystyle \forall {is}\,\,{\mbox{isométrie de}}\,\,\mathbb {R} ^{n}}
c
a
r
d
Q
,
R
(
i
s
(
A
)
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{is}(A){\Big )}={card}_{Q,{\cal {R}}}(A)}
∀
A
∈
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle \forall A\in {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
∀
x
∈
R
n
,
c
a
r
d
Q
,
R
(
t
x
(
A
)
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle \displaystyle {\forall x\in \mathbb {R} ^{n},\,\,{card}_{Q,{\cal {R}}}{\Big (}t_{x}(A){\Big )}={card}_{Q,{\cal {R}}}(A)}}
où
∀
x
∈
R
n
,
t
x
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
A
+
x
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,t_{x}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,A+x}
x
{\displaystyle x}
R
n
{\displaystyle \mathbb {R} ^{n}}
donc, en particulier,
∀
A
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle \forall A\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
x
∈
R
n
,
c
a
r
d
Q
,
R
(
t
x
(
A
)
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle \displaystyle {\forall x\in \mathbb {R} ^{n},\,\,{card}_{Q,{\cal {R}}}{\Big (}t_{x}(A){\Big )}={card}_{Q,{\cal {R}}}(A)}}
où
∀
x
∈
R
n
,
t
x
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
A
+
x
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,t_{x}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,A+x}
x
{\displaystyle x}
R
n
{\displaystyle \mathbb {R} ^{n}}
a2)
∀
A
∈
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle \forall A\in {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
∀
M
∈
R
n
{\displaystyle \forall M\in \mathbb {R} ^{n}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
c
a
r
d
Q
,
R
(
r
o
t
(
M
,
θ
n
)
(
A
)
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{rot}_{(M,\theta _{n})}(A){\Big )}={card}_{Q,{\cal {R}}}(A)}
où
∀
M
∈
R
n
{\displaystyle \displaystyle {\forall M\in \mathbb {R} ^{n}}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
r
o
t
(
M
,
θ
n
)
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
r
o
t
(
M
,
θ
n
)
(
A
)
{\displaystyle {rot}_{(M,\theta _{n})}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,{rot}_{(M,\theta _{n})}(A)}
M
{\displaystyle M}
θ
n
{\displaystyle \theta _{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
donc, en particulier,
∀
A
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle \forall A\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
M
∈
R
n
{\displaystyle \forall M\in \mathbb {R} ^{n}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
c
a
r
d
Q
,
R
(
r
o
t
(
M
,
θ
n
)
(
A
)
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}{rot}_{(M,\theta _{n})}(A){\Big )}={card}_{Q,{\cal {R}}}(A)}
où
∀
M
∈
R
n
{\displaystyle \displaystyle {\forall M\in \mathbb {R} ^{n}}}
∀
θ
n
∈
{
{
0
,
π
}
+
2
π
Z
si
n
=
1
R
n
−
1
si
n
≠
1
{\displaystyle \forall \theta _{n}\in {\begin{cases}\{0,\pi \}+2\pi \mathbb {Z} &{\text{si }}n=1\\{\mathbb {R} }^{n-1}&{\text{si }}n\neq 1\\\end{cases}}}
r
o
t
(
M
,
θ
n
)
:
P
(
R
n
)
→
P
(
R
n
)
:
A
↦
r
o
t
(
M
,
θ
n
)
(
A
)
{\displaystyle {rot}_{(M,\theta _{n})}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,\rightarrow \,\,{\mathcal {P}}(\mathbb {R} ^{n})\,\,:\,\,A\,\,\mapsto \,\,{rot}_{(M,\theta _{n})}(A)}
M
{\displaystyle M}
θ
n
{\displaystyle \theta _{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
Fin du théorème
Remarque :
On verra que
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
(
v
o
l
i
)
i
∈
N
n
{\displaystyle {({vol}^{i})}_{i\in \mathbb {N} _{n}}}
i
(
i
∈
N
n
)
{\displaystyle i\,\,(i\in \mathbb {N} _{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
v
o
l
0
{\displaystyle {vol}^{0}}
R
n
{\displaystyle \mathbb {R} ^{n}}
dans La saga du "cardinal" version 4 (voir supra
ou dans : Théorème (
c
a
r
d
Q
~
∈
C
0
(
P
V
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}\in {\mathcal {C}}^{0}{\Big (}{PV}(\mathbb {R} ^{N}),F{\Big )}}}
c
a
r
d
Q
,
i
~
∈
C
0
(
P
V
i
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,i}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{i}(\mathbb {R} ^{N}),F{\Big )}}}
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
A
N
=
A
N
N
∈
P
V
N
(
R
N
)
{\displaystyle A_{N}=A_{N}^{N}\in {PV}_{N}(\mathbb {R} ^{N})}
[
0
,
1
[
{\displaystyle [0,1[}
ou dans les propositions suivantes : Proposition 1.4 de Guillaume FOUCART, dans les PDF de Michel COSTE (voir infra et Proposition (voir infra
Le problème de cette définition est que l'ensemble d'arrivée dépend de
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
Quant à l'introduction de l'anneau commutatif unitaire intègre ordonné
(
F
,
+
,
×
,
≤
)
{\displaystyle (F,+,\times ,\leq )}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
c
a
r
d
Q
,
R
(
P
(
R
n
)
)
{\displaystyle {card}_{Q,{\mathcal {R}}}{\Big (}{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
P
(
R
n
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
N
⨆
+
∞
{\displaystyle \displaystyle {\mathbb {N} \bigsqcup +\infty }}
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle +\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
.
Denis FELDMANN m'a mis au courant du paradoxe de BURALI-FORTI que je vais rencontrer, concernant l'ensemble
+
∞
{\displaystyle +\infty }
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
N
⨆
+
∞
{\displaystyle \mathbb {N} \bigsqcup +\infty }
Remarque importante : Obstacle et facteur, pour l'instant, limitant de "ma théorie":
Dans le cas des parties de
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
P
(
R
n
)
{\displaystyle {\mathcal {P}}({\mathbb {R} }^{n})}
″
[
A
,
(
A
i
)
i
∈
I
]
″
{\displaystyle ''[A,{(A_{i})}_{i\in I}]''}
A
∈
P
V
2
(
R
n
)
{\displaystyle A\in {PV2}(\mathbb {R} ^{n})}
(
A
i
)
i
∈
I
⊂
P
V
(
R
n
)
{\displaystyle {(A_{i})}_{i\in I}\subset {PV}(\mathbb {R} ^{n})}
Remarque importante :
B
{\displaystyle B}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
au lieu de parler de la F-quantité relative au repère
R
{\displaystyle {\mathcal {R}}}
B
{\displaystyle B}
c
a
r
d
Q
,
R
(
B
)
{\displaystyle {card}_{Q,{\mathcal {R}}}(B)}
R
{\displaystyle {\mathcal {R}}}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
B
{\displaystyle B}
c
a
r
d
Q
,
R
,
[
A
,
(
A
i
)
i
∈
I
]
(
B
)
{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(B)}
et dans ce cas on a : "
c
a
r
d
Q
,
R
,
[
A
,
(
A
i
)
i
∈
I
]
(
B
)
=
d
e
´
f
c
a
r
d
Q
,
R
(
B
⋂
[
A
,
(
A
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
B
⋂
A
,
(
B
⋂
A
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(B){\underset {d{\acute {e}}f}{=}}{card}_{Q,{\mathcal {R}}}{\Big (}B\bigcap [A,{(A_{i})}_{i\in I}]{\Big )}={card}_{Q,{\mathcal {R}}}{\Big (}{\Big [}B\bigcap A,{{\Big (}B\bigcap A_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}}
et, en particulier, on a : "
c
a
r
d
Q
,
R
,
[
A
,
(
A
i
)
i
∈
I
]
(
A
)
=
d
e
´
f
c
a
r
d
Q
,
R
(
A
⋂
[
A
,
(
A
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
⋂
A
i
)
i
∈
I
]
)
=
p
r
o
p
?
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(A){\underset {d{\acute {e}}f}{=}}{card}_{Q,{\mathcal {R}}}{\Big (}A\bigcap [A,{(A_{i})}_{i\in I}]{\Big )}={card}_{Q,{\mathcal {R}}}{\Big (}{\Big [}A,{{\Big (}A\bigcap A_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}{\underset {prop?}{=}}{card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])}
Quand on parle de "
c
a
r
d
Q
,
R
,
[
A
,
(
A
i
)
i
∈
I
]
(
B
)
{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(B)}
R
{\displaystyle {\mathcal {R}}}
Lorsque la famille
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
C
1
{\displaystyle {\mathcal {C}}^{1}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])}
R
{\displaystyle {\mathcal {R}}}
Problème important (lignes ajoutées le 29/05/2021) :
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
Le fait de remplacer le terme "convexe" par celui de "polyconvexe" (et donc le terme "connexe" par le terme "non connexe" ou rien du tout), dans la définition de
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
Peut-être que pour créer la tribu adéquate que l'on souhaite, il faut ajouter aux parties de
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
Mais, alors il faut parler de la F-quantité de
R
n
{\displaystyle \mathbb {R} ^{n}}
[
R
n
,
(
A
i
)
i
∈
I
]
{\displaystyle [\mathbb {R} ^{n},{(A_{i})}_{i\in I}]}
P
(
R
n
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})}
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Il en découle de 1)b), de 2) et peut-être d'autres hypothèses de définition de la F-quantité, en particulier que :
∀
I
∈
P
(
R
n
)
,
c
a
r
d
P
(
I
)
≤
ℵ
0
,
∀
(
A
i
)
i
∈
I
⊂
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle \forall I\in {\cal {P}}({\mathbb {R} }^{n}),\,\,{card}_{P}(I)\leq \aleph _{0},\,\,\forall (A_{i})_{i\in I}\subset {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
i
,
j
∈
I
,
i
≠
j
,
A
i
⋂
A
j
=
∅
{\displaystyle \forall i,j\in I,\,\,i\neq j,\,\,A_{i}\bigcap A_{j}=\emptyset }
c
a
r
d
Q
,
R
(
⨆
i
∈
I
A
i
)
=
∑
i
∈
I
c
a
r
d
Q
,
R
(
A
i
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in I}A_{i}{\Big )}=\sum _{i\in I}{card}_{Q,{\cal {R}}}(A_{i})}
Plus tard, avec les outils dont nous disposerons, nous pourrions même montrer que :
∀
I
∈
P
(
R
n
)
,
c
a
r
d
P
(
I
)
≤
ℵ
0
,
∀
(
A
i
)
i
∈
I
⊂
P
V
(
R
n
)
{\displaystyle \forall I\in {\cal {P}}({\mathbb {R} }^{n}),\,\,{card}_{P}(I)\leq \aleph _{0},\,\,\forall (A_{i})_{i\in I}\subset {PV}(\mathbb {R} ^{n})}
∀
i
,
j
∈
I
,
i
≠
j
,
A
i
⋂
A
j
=
∅
{\displaystyle \forall i,j\in I,\,\,i\neq j,\,\,A_{i}\bigcap A_{j}=\emptyset }
c
a
r
d
Q
,
R
(
⨆
i
∈
I
A
i
)
=
∑
i
∈
I
c
a
r
d
Q
,
R
(
A
i
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in I}A_{i}{\Big )}=\sum _{i\in I}{card}_{Q,{\cal {R}}}(A_{i})}
La
σ
{\displaystyle \sigma }
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
(Cf. définition de
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
∀
A
,
B
∈
P
(
R
n
)
,
B
∈
P
(
A
)
{\displaystyle \forall A,B\in {\mathcal {P}}(\mathbb {R} ^{n}),\,\,B\in {\mathcal {P}}(A)}
c
a
r
d
Q
,
R
(
A
∖
B
)
=
c
a
r
d
Q
,
R
(
A
)
−
c
a
r
d
Q
,
R
(
B
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(A\setminus B)={card}_{Q,{\cal {R}}}(A)-{card}_{Q,{\cal {R}}}(B)}}
∀
A
,
B
∈
P
(
R
n
)
,
A
∈
P
(
B
)
,
A
≠
B
{\displaystyle \forall A,B\in {\mathcal {P}}(\mathbb {R} ^{n}),\,\,A\in {\mathcal {P}}(B),\,\,A\neq B}
c
a
r
d
Q
,
R
(
A
)
<
c
a
r
d
Q
,
R
(
B
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(A)<{card}_{Q,{\cal {R}}}(B)}}
I
,
(
I
i
)
i
∈
N
n
{\displaystyle I,{(I_{i})}_{i\in \mathbb {N} _{n}}}
(
I
,
(
I
i
)
i
∈
N
n
)
⊂
P
o
l
y
t
o
p
e
s
(
R
)
{\displaystyle {\Big (}I,{(I_{i})}_{i\in \mathbb {N} _{n}}{\Big )}\subset {{\mathcal {P}}olytopes}(\mathbb {R} )}
R
{\displaystyle \mathbb {R} }
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\mathcal {R}}}}
c
a
r
d
Q
,
R
(
∏
i
∈
N
n
∗
I
i
)
=
c
a
r
d
Q
,
R
n
(
∏
i
∈
N
n
∗
I
i
)
=
∏
i
∈
N
n
∗
c
a
r
d
Q
,
R
1
(
I
i
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(\prod _{i\in {\mathbb {N} }_{n}^{*}}I_{i})={card}_{Q,{\cal {R}}_{n}}(\prod _{i\in {\mathbb {N} }_{n}^{*}}I_{i})=\prod _{i\in {\mathbb {N} }_{n}^{*}}{card}_{Q,{\cal {R}}_{1}}(I_{i})}}
c
a
r
d
Q
,
R
(
I
n
)
=
c
a
r
d
Q
,
R
n
(
I
n
)
=
c
a
r
d
Q
,
R
n
(
∏
i
∈
N
n
∗
I
)
=
∏
i
∈
N
n
∗
c
a
r
d
Q
,
R
1
(
I
)
=
(
c
a
r
d
Q
,
R
1
(
I
)
)
n
=
c
a
r
d
Q
,
R
1
n
(
I
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(I^{n})={card}_{Q,{\cal {R}}_{n}}(I^{n})={card}_{Q,{\cal {R}}_{n}}(\prod _{i\in {\mathbb {N} }_{n}^{*}}I)=\prod _{i\in {\mathbb {N} }_{n}^{*}}{card}_{Q,{\cal {R}}_{1}}(I)={{\Big (}{card}_{Q,{\cal {R}}_{1}}(I){\Big )}}^{n}={card}_{Q,{\cal {R}}_{1}}^{n}(I)}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
∀
x
∈
R
n
,
c
a
r
d
Q
,
R
(
{
x
}
)
=
1
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,{card}_{Q,{\cal {R}}}(\{x\})=1}
Proposition :
E
~
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle {\widetilde {E}}\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
Si
∀
x
∈
E
~
,
A
x
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle \forall x\in {\widetilde {E}},\,\,A_{x}\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
∀
x
,
y
∈
E
~
,
x
≠
y
,
A
x
⋂
A
y
=
∅
{\displaystyle \forall x,y\in {\widetilde {E}},\,\,x\neq y,\,\,A_{x}\bigcap A_{y}=\emptyset }
A
=
⨆
x
∈
E
~
A
x
{\displaystyle A=\bigsqcup _{x\in {\widetilde {E}}}A_{x}}
alors
c
a
r
d
Q
,
R
(
A
)
=
c
a
r
d
Q
,
R
(
⨆
x
∈
E
~
A
x
)
=
∫
E
~
c
a
r
d
Q
,
R
(
A
x
)
d
c
a
r
d
Q
,
R
(
x
)
{\displaystyle {card}_{Q,{\cal {R}}}(A)={card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{x\in {\widetilde {E}}}A_{x}{\Big )}=\int _{\widetilde {E}}{card}_{Q,{\cal {R}}}(A_{x})\,\,d\,\,{card}_{Q,{\cal {R}}}(x)}
Plus tard, avec les outils dont nous disposerons, nous pourrions même montrer que :
Soit
E
~
∈
P
V
(
R
n
)
{\displaystyle {\widetilde {E}}\in {PV}(\mathbb {R} ^{n})}
Si
∀
x
∈
E
~
,
A
x
∈
P
V
(
R
n
)
{\displaystyle \forall x\in {\widetilde {E}},\,\,A_{x}\in {PV}(\mathbb {R} ^{n})}
∀
x
,
y
∈
E
~
,
x
≠
y
,
A
x
⋂
A
y
=
∅
{\displaystyle \forall x,y\in {\widetilde {E}},\,\,x\neq y,\,\,A_{x}\bigcap A_{y}=\emptyset }
A
=
⨆
x
∈
E
~
A
x
{\displaystyle A=\bigsqcup _{x\in {\widetilde {E}}}A_{x}}
alors
c
a
r
d
Q
,
R
(
A
)
=
c
a
r
d
Q
,
R
(
⨆
x
∈
E
~
A
x
)
=
∫
E
~
c
a
r
d
Q
,
R
(
A
x
)
d
c
a
r
d
Q
,
R
(
x
)
{\displaystyle {card}_{Q,{\cal {R}}}(A)={card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{x\in {\widetilde {E}}}A_{x}{\Big )}=\int _{\widetilde {E}}{card}_{Q,{\cal {R}}}(A_{x})\,\,d\,\,{card}_{Q,{\cal {R}}}(x)}
(sous réserve de conditions supplémentaires concernant les parties de
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
E
~
∈
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle {\widetilde {E}}\in {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
E
~
∈
P
V
(
R
n
)
{\displaystyle {\widetilde {E}}\in {PV}(\mathbb {R} ^{n})}
E
~
∈
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle {\widetilde {E}}\in {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
(Formule peut-être remise en cause car la notion de F-quantité n'est pas a priori une mesure définie sur
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
P
o
l
y
t
o
p
e
s
(
R
n
)
{\displaystyle {{\mathcal {P}}olytopes}(\mathbb {R} ^{n})}
Fin du théorème
Existence et résultats sur les intervalles
I
{\displaystyle I}
R
{\displaystyle \mathbb {R} }
P
V
(
R
)
{\displaystyle {PV}(\mathbb {R} )}
[ modifier | modifier le wikicode ] Soit
R
{\displaystyle {\cal {R}}}
R
{\displaystyle \mathbb {R} }
O
{\displaystyle O}
Préliminaires :
Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
A
∈
P
(
E
)
{\displaystyle A\in {\cal {P}}(E)}
A
∘
E
{\displaystyle {\stackrel {\circ }{A}}^{E}}
A
{\displaystyle A}
|
{\displaystyle |}
E
{\displaystyle E}
A
∘
{\displaystyle {\stackrel {\circ }{A}}}
A
¯
E
{\displaystyle {\overline {A}}^{E}}
A
{\displaystyle A}
|
{\displaystyle |}
E
{\displaystyle E}
A
¯
{\displaystyle {\overline {A}}}
∀
i
∈
N
N
∗
,
∀
n
∈
N
N
∗
,
n
≥
i
,
v
o
l
i
,
n
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N}^{*},\,\,\forall n\in \mathbb {N} _{N}^{*},\,\,n\geq i,\,\,{vol}^{i,n}}}
i
{\displaystyle i}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
i
,
n
=
{
A
i
,
n
∈
B
(
R
n
)
|
d
i
m
(
A
i
,
n
)
=
i
}
⋃
{
∅
}
{\displaystyle \displaystyle {{\cal {A}}_{i,n}=\{A_{i,n}\in {\cal {B}}(\mathbb {R} ^{n})|{dim}(A_{i,n})=i\}\bigcup \{\emptyset \}}}
v
o
l
i
,
n
{\displaystyle {vol}^{i,n}}
v
o
l
i
{\displaystyle {vol}^{i}}
∀
n
∈
N
N
∗
,
v
o
l
0
,
n
{\displaystyle \displaystyle {\forall n\in \mathbb {N} _{N}^{*},\,\,{vol}^{0,n}}}
0
{\displaystyle 0}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
0
,
n
=
{
A
0
,
n
∈
P
(
R
n
)
|
d
i
m
(
A
0
,
n
)
=
0
}
⋃
{
∅
}
=
{
A
0
,
n
∈
P
(
R
n
)
|
c
a
r
d
P
(
A
0
,
n
)
≤
ℵ
0
}
{\displaystyle \displaystyle {{\cal {A}}_{0,n}=\{A_{0,n}\in {\cal {P}}(\mathbb {R} ^{n})|{dim}(A_{0,n})=0\}\bigcup \{\emptyset \}=\{A_{0,n}\in {\cal {P}}(\mathbb {R} ^{n})|{card}_{P}(A_{0,n})\leq \aleph _{0}\}}}
v
o
l
0
,
n
{\displaystyle {vol}^{0,n}}
v
o
l
0
{\displaystyle {vol}^{0}}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
Soit
n
∈
N
N
∗
,
n
≥
i
{\displaystyle n\in \mathbb {N} _{N}^{*},\,\,n\geq i}
v
o
l
i
~
{\displaystyle {\widetilde {{vol}^{i}}}}
v
o
l
i
{\displaystyle {vol}^{i}}
i
{\displaystyle i}
R
n
{\displaystyle {\mathbb {R} }^{n}}
v
o
l
i
,
n
{\displaystyle {vol}^{i,n}}
⨆
n
∈
N
N
∗
,
n
≥
i
R
n
{\displaystyle \displaystyle {\bigsqcup _{n\in \mathbb {N} _{N}^{*},\,\,n\geq i}{\mathbb {R} }^{n}}}
A
i
=
⨆
n
∈
N
N
∗
,
n
≥
i
A
i
,
n
{\displaystyle \displaystyle {{\cal {A}}_{i}=\bigsqcup _{n\in \mathbb {N} _{N}^{*},\,\,n\geq i}{\cal {A}}_{i,n}}}
∀
n
∈
N
N
∗
,
n
≥
i
,
v
o
l
i
~
|
A
i
,
n
=
v
o
l
i
,
n
{\displaystyle \forall n\in \mathbb {N} _{N}^{*},\,\,n\geq i,\,\,{\widetilde {{vol}^{i}}}_{|{\cal {A}}_{i,n}}={vol}^{i,n}}
et telle que
∀
n
∈
N
N
∗
,
n
≥
i
,
∃
A
i
,
n
∈
A
i
,
n
,
A
i
=
⨆
n
∈
N
N
∗
,
n
≥
i
A
i
,
n
,
v
o
l
i
~
(
A
i
)
=
v
o
l
i
~
(
⨆
n
∈
N
N
∗
,
n
≥
i
A
i
,
n
)
=
∑
n
∈
N
N
∗
,
n
≥
i
v
o
l
i
~
(
A
i
,
n
)
{\displaystyle \displaystyle {\forall n\in \mathbb {N} _{N}^{*},\,\,n\geq i,\,\,\exists A_{i,n}\in {\cal {A}}_{i,n},\,\,A_{i}=\bigsqcup _{n\in \mathbb {N} _{N}^{*},n\geq i}A_{i,n},\,\,{\widetilde {{vol}^{i}}}(A_{i})={\widetilde {{vol}^{i}}}(\bigsqcup _{n\in \mathbb {N} _{N}^{*},\,\,n\geq i}A_{i,n})=\sum _{n\in \mathbb {N} _{N}^{*},\,\,n\geq i}{\widetilde {{vol}^{i}}}(A_{i,n})}}
Fin du théorème
Début d’un théorème
Soient
I
{\displaystyle I}
J
{\displaystyle J}
R
{\displaystyle \mathbb {R} }
I
¯
{\displaystyle {\overline {I}}}
J
¯
{\displaystyle {\overline {J}}}
I
∘
{\displaystyle {\stackrel {\circ }{I}}}
J
∘
{\displaystyle {\stackrel {\circ }{J}}}
i
0
{\displaystyle i_{0}}
j
0
{\displaystyle j_{0}}
1)
∀
x
0
∈
R
n
,
c
a
r
d
Q
,
R
(
{
x
0
}
)
=
v
o
l
0
(
{
x
0
}
)
=
1
{\displaystyle \displaystyle {\forall x_{0}\in \mathbb {R} ^{n},\,\,{card}_{Q,{\cal {R}}}(\{x_{0}\})={vol}^{0}(\{x_{0}\})=1}}
v
o
l
1
(
I
)
=
v
o
l
1
(
J
)
⟺
c
a
r
d
Q
,
R
(
I
¯
)
=
c
a
r
d
Q
,
R
(
J
¯
)
e
t
c
a
r
d
Q
,
R
(
I
∘
)
=
c
a
r
d
Q
,
R
(
J
∘
)
{\displaystyle \displaystyle {{vol}^{1}(I)={vol}^{1}(J)\Longleftrightarrow {card}_{Q,{\cal {R}}}({\overline {I}})={card}_{Q,{\cal {R}}}({\overline {J}})\,\,et\,\,{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})={card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})}}
En effet
∀
I
,
J
i
n
t
e
r
v
a
l
l
e
b
o
r
n
e
´
d
e
R
:
v
o
l
1
(
I
)
=
v
o
l
1
(
J
)
,
∃
a
I
,
J
∈
R
,
J
¯
=
I
¯
+
a
I
,
J
e
t
J
∘
=
I
∘
+
a
I
,
J
{\displaystyle \displaystyle {\forall I,J\,\,intervalle\,\,born{\acute {e}}\,\,de\,\,\mathbb {R} \,\,:\,\,{vol}^{1}(I)={vol}^{1}(J),\,\,\exists a_{I,J}\in \mathbb {R} ,\,\,{\overline {J}}={\overline {I}}+a_{I,J}\,\,et\,\,{\stackrel {\circ }{J}}={\stackrel {\circ }{I}}+a_{I,J}}}
2)
c
a
r
d
Q
,
R
(
I
¯
∖
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
¯
∖
{
j
0
}
)
=
c
a
r
d
Q
,
R
(
(
I
∘
⨆
∂
I
)
∖
{
i
0
}
)
c
a
r
d
Q
,
R
(
(
J
∘
⨆
∂
J
)
∖
{
j
0
}
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}}\setminus \{i_{0}\})}{{card}_{Q,{\cal {R}}}({\overline {J}}\setminus \{j_{0}\})}}={\frac {{card}_{Q,{\cal {R}}}{\Big (}({\stackrel {\circ }{I}}\bigsqcup \partial I)\setminus \{i_{0}\}{\Big )}}{{card}_{Q,{\cal {R}}}{\Big (}({\stackrel {\circ }{J}}\bigsqcup \partial J)\setminus \{j_{0}\}{\Big )}}}}}
c'est-à-dire
c
a
r
d
Q
,
R
(
I
¯
)
−
c
a
r
d
Q
,
R
(
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
¯
)
−
c
a
r
d
Q
,
R
(
{
j
0
}
)
=
c
a
r
d
Q
,
R
(
I
∘
⨆
∂
I
)
−
c
a
r
d
Q
,
R
(
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
∘
⨆
∂
J
)
−
c
a
r
d
Q
,
R
(
{
j
0
}
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-{card}_{Q,{\cal {R}}}(\{i_{0}\})}{{card}_{Q,{\cal {R}}}({\overline {J}})-{card}_{Q,{\cal {R}}}(\{j_{0}\})}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}}\bigsqcup \partial I)-{card}_{Q,{\cal {R}}}(\{i_{0}\})}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}}\bigsqcup \partial J)-{card}_{Q,{\cal {R}}}(\{j_{0}\})}}}}
c'est-à-dire
c
a
r
d
Q
,
R
(
I
¯
)
−
c
a
r
d
Q
,
R
(
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
¯
)
−
c
a
r
d
Q
,
R
(
{
j
0
}
)
=
c
a
r
d
Q
,
R
(
I
∘
)
+
c
a
r
d
Q
,
R
(
∂
I
)
−
c
a
r
d
Q
,
R
(
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
∘
)
+
c
a
r
d
Q
,
R
(
∂
J
)
−
c
a
r
d
Q
,
R
(
{
j
0
}
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-{card}_{Q,{\cal {R}}}(\{i_{0}\})}{{card}_{Q,{\cal {R}}}({\overline {J}})-{card}_{Q,{\cal {R}}}(\{j_{0}\})}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+{card}_{Q,{\cal {R}}}(\partial I)-{card}_{Q,{\cal {R}}}(\{i_{0}\})}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+{card}_{Q,{\cal {R}}}(\partial J)-{card}_{Q,{\cal {R}}}(\{j_{0}\})}}}}
c'est-à-dire
c
a
r
d
Q
,
R
(
I
¯
)
−
c
a
r
d
Q
,
R
(
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
¯
)
−
c
a
r
d
Q
,
R
(
{
j
0
}
)
=
c
a
r
d
Q
,
R
(
I
∘
)
+
c
a
r
d
Q
,
R
(
{
inf
i
∈
I
i
,
sup
i
∈
I
i
}
)
−
c
a
r
d
Q
,
R
(
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
∘
)
+
c
a
r
d
Q
,
R
(
{
inf
j
∈
J
j
,
sup
j
∈
J
j
}
)
−
c
a
r
d
Q
,
R
(
{
j
0
}
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-{card}_{Q,{\cal {R}}}(\{i_{0}\})}{{card}_{Q,{\cal {R}}}({\overline {J}})-{card}_{Q,{\cal {R}}}(\{j_{0}\})}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+{card}_{Q,{\cal {R}}}(\{\displaystyle {\inf _{i\in I}i,\sup _{i\in I}i}\})-{card}_{Q,{\cal {R}}}(\{i_{0}\})}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+{card}_{Q,{\cal {R}}}(\{\displaystyle {\inf _{j\in J}j,\sup _{j\in J}j}\})-{card}_{Q,{\cal {R}}}(\{j_{0}\})}}}}
c'est-à-dire
c
a
r
d
Q
,
R
(
I
¯
)
−
1
c
a
r
d
Q
,
R
(
J
¯
)
−
1
=
c
a
r
d
Q
,
R
(
I
∘
)
+
2
−
1
c
a
r
d
Q
,
R
(
J
∘
)
+
2
−
1
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-1}{{card}_{Q,{\cal {R}}}({\overline {J}})-1}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+2-1}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+2-1}}}}
c'est-à-dire
c
a
r
d
Q
,
R
(
I
¯
)
−
1
c
a
r
d
Q
,
R
(
J
¯
)
−
1
=
c
a
r
d
Q
,
R
(
I
∘
)
+
1
c
a
r
d
Q
,
R
(
J
∘
)
+
1
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-1}{{card}_{Q,{\cal {R}}}({\overline {J}})-1}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+1}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+1}}}}
Fin du théorème
[ modifier | modifier le wikicode ] Début d’un théorème
Soient
I
{\displaystyle I}
J
{\displaystyle J}
R
{\displaystyle \mathbb {R} }
I
¯
{\displaystyle {\overline {I}}}
J
¯
{\displaystyle {\overline {J}}}
I
∘
{\displaystyle {\stackrel {\circ }{I}}}
J
∘
{\displaystyle {\stackrel {\circ }{J}}}
i
0
{\displaystyle i_{0}}
j
0
{\displaystyle j_{0}}
c
a
r
d
Q
,
R
(
I
¯
∖
{
i
0
}
)
c
a
r
d
Q
,
R
(
J
¯
∖
{
j
0
}
)
=
c
a
r
d
Q
,
R
(
I
¯
)
−
1
c
a
r
d
Q
,
R
(
J
¯
)
−
1
=
c
a
r
d
Q
,
R
(
I
∘
)
+
1
c
a
r
d
Q
,
R
(
J
∘
)
+
1
=
v
o
l
1
(
I
)
v
o
l
1
(
J
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}}\setminus \{i_{0}\})}{{card}_{Q,{\cal {R}}}({\overline {J}}\setminus \{j_{0}\})}}={\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-1}{{card}_{Q,{\cal {R}}}({\overline {J}})-1}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+1}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+1}}}={\frac {vol^{1}(I)}{vol^{1}(J)}}}
Fin du théorème
Démonstration :
I
{\displaystyle I}
J
{\displaystyle J}
R
{\displaystyle \mathbb {R} }
R
{\displaystyle \mathbb {R} }
K
0
,
I
∘
=
]
0
,
v
o
l
1
(
I
∘
)
[
{\displaystyle \displaystyle {K_{0,{\stackrel {\circ }{I}}}=]0,{vol}^{1}({\stackrel {\circ }{I}})[}}
K
0
,
J
∘
=
]
0
,
v
o
l
1
(
J
∘
)
[
{\displaystyle \displaystyle {K_{0,{\stackrel {\circ }{J}}}=]0,{vol}^{1}({\stackrel {\circ }{J}})[}}
K
0
,
I
¯
=
[
0
,
v
o
l
1
(
I
¯
)
]
{\displaystyle \displaystyle {K_{0,{\overline {I}}}=[0,{vol}^{1}({\overline {I}})]}}
K
0
,
J
¯
=
[
0
,
v
o
l
1
(
J
¯
)
]
{\displaystyle \displaystyle {K_{0,{\overline {J}}}=[0,{vol}^{1}({\overline {J}})]}}
c
a
r
d
Q
,
R
(
K
0
,
I
∘
)
+
1
c
a
r
d
Q
,
R
(
K
0
,
J
∘
)
+
1
=
v
o
l
1
(
K
0
,
I
∘
)
v
o
l
1
(
K
0
,
J
∘
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\mathcal {R}}}(K_{0,{\stackrel {\circ }{I}}})+1}{{card}_{Q,{\mathcal {R}}}(K_{0,{\stackrel {\circ }{J}}})+1}}={\frac {{vol}^{1}(K_{0,{\stackrel {\circ }{I}}})}{{vol}^{1}(K_{0,{\stackrel {\circ }{J}}})}}}}
En effet,on a (proposition):
s
=
k
∈
N
{\displaystyle s=k\in \mathbb {N} }
[
0
,
k
p
[
=
⨆
i
∈
N
k
∗
[
(
i
−
1
)
p
,
i
p
[
{\displaystyle \displaystyle {[0,kp[=\bigsqcup _{i\in \mathbb {N} _{k}^{*}}[(i-1)p,ip[}}
donc
c
a
r
d
Q
,
R
(
[
0
,
k
p
[
)
=
∑
i
∈
N
k
∗
c
a
r
d
Q
,
R
(
[
(
i
−
1
)
p
,
i
p
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([0,kp[)=\sum _{i\in \mathbb {N} _{k}^{*}}{card}_{Q,{\cal {R}}}([(i-1)p,ip[)}}
∀
i
∈
N
k
∗
,
c
a
r
d
Q
,
R
(
]
(
i
−
1
)
p
,
i
p
[
)
=
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{k}^{*},\,\,{card}_{Q,{\cal {R}}}(](i-1)p,ip[)={card}_{Q,{\cal {R}}}(]0,p[)}}
car
∀
i
∈
N
k
∗
,
v
o
l
1
(
]
(
i
−
1
)
p
,
i
p
[
)
=
v
o
l
1
(
]
0
,
p
[
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{k}^{*},\,\,{vol}^{1}(](i-1)p,ip[)={vol}^{1}(]0,p[)}}
∀
i
∈
N
k
∗
,
c
a
r
d
Q
,
R
(
]
(
i
−
1
)
p
,
i
p
[
)
+
1
=
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
1
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{k}^{*},\,\,{card}_{Q,{\cal {R}}}(](i-1)p,ip[)+1={card}_{Q,{\cal {R}}}(]0,p[)+1}}
donc
∀
i
∈
N
k
∗
,
c
a
r
d
Q
,
R
(
]
(
i
−
1
)
p
,
i
p
[
)
+
c
a
r
d
Q
,
R
(
{
(
i
−
1
)
p
}
)
=
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{k}^{*},\,\,{card}_{Q,{\cal {R}}}(](i-1)p,ip[)+{card}_{Q,{\cal {R}}}(\{(i-1)p\})={card}_{Q,{\cal {R}}}(]0,p[)+{card}_{Q,{\cal {R}}}(\{0\})}}
donc
∀
i
∈
N
k
∗
,
c
a
r
d
Q
,
R
(
]
(
i
−
1
)
p
,
i
p
[
⨆
{
(
i
−
1
)
p
}
)
=
c
a
r
d
Q
,
R
(
]
0
,
p
[
⨆
{
0
}
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{k}^{*},\,\,{card}_{Q,{\cal {R}}}(](i-1)p,ip[\bigsqcup \{(i-1)p\})={card}_{Q,{\cal {R}}}(]0,p[\bigsqcup \{0\})}}
donc
∀
i
∈
N
k
∗
,
c
a
r
d
Q
,
R
(
[
(
i
−
1
)
p
,
i
p
[
)
=
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{k}^{*},\,\,{card}_{Q,{\cal {R}}}([(i-1)p,ip[)={card}_{Q,{\cal {R}}}([0,p[)}}
c
a
r
d
Q
,
R
(
[
0
,
k
p
[
)
=
∑
i
∈
N
k
∗
c
a
r
d
Q
,
R
(
[
(
i
−
1
)
p
,
i
p
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([0,kp[)=\sum _{i\in \mathbb {N} _{k}^{*}}{card}_{Q,{\cal {R}}}([(i-1)p,ip[)}}
c
a
r
d
Q
,
R
(
[
0
,
k
p
[
)
=
∑
i
∈
N
k
∗
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([0,kp[)=\sum _{i\in \mathbb {N} _{k}^{*}}{card}_{Q,{\cal {R}}}([0,p[)}}
donc
c
a
r
d
Q
,
R
(
[
0
,
k
p
[
)
=
k
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([0,kp[)=k\,{card}_{Q,{\cal {R}}}([0,p[)}}
donc
c
a
r
d
Q
,
R
(
]
0
,
k
p
[
)
=
c
a
r
d
Q
,
R
(
[
0
,
k
p
[
∖
{
0
}
)
=
c
a
r
d
Q
,
R
(
[
0
,
k
p
[
)
−
c
a
r
d
Q
,
R
(
{
0
}
)
=
k
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
−
1
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(]0,kp[)={card}_{Q,{\cal {R}}}([0,kp[\setminus \{0\})={card}_{Q,{\cal {R}}}([0,kp[)-{card}_{Q,{\cal {R}}}(\{0\})=k\,\,{card}_{Q,{\cal {R}}}([0,p[)-1}}
=
k
(
c
a
r
d
Q
,
R
(
]
0
,
p
[
⨆
{
0
}
)
)
−
1
=
k
(
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
)
−
1
=
k
(
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
1
)
−
1
{\displaystyle \displaystyle {=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[\bigsqcup \{0\}){\Big )}-1=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[)+{card}_{Q,{\cal {R}}}(\{0\}){\Big )}-1=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[)+1{\Big )}-1}}
•(2)
]
0
,
k
p
[
=
(
⨆
i
∈
N
k
∗
]
(
i
−
1
)
p
,
i
p
[
)
⨆
(
⨆
i
∈
N
k
−
1
∗
{
i
p
}
)
{\displaystyle \displaystyle {]0,kp[=(\bigsqcup _{i\in \mathbb {N} _{k}^{*}}](i-1)p,ip[)\bigsqcup (\bigsqcup _{i\in \mathbb {N} _{k-1}^{*}}\{ip\})}}
donc
c
a
r
d
Q
,
R
(
]
0
,
k
p
[
)
=
∑
i
∈
N
k
∗
c
a
r
d
Q
,
R
(
]
(
i
−
1
)
p
,
i
p
[
)
+
∑
i
∈
N
k
−
1
∗
c
a
r
d
Q
,
R
(
{
i
p
}
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(]0,kp[)=\sum _{i\in \mathbb {N} _{k}^{*}}{card}_{Q,{\mathcal {R}}}(](i-1)p,ip[)+\sum _{i\in \mathbb {N} _{k-1}^{*}}{card}_{Q,{\mathcal {R}}}(\{ip\})}}
∀
i
∈
N
k
∗
,
c
a
r
d
Q
,
R
(
]
(
i
−
1
)
p
,
i
p
[
)
=
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
{\displaystyle \forall i\in \mathbb {N} _{k}^{*},\,\,{card}_{Q,{\mathcal {R}}}(](i-1)p,ip[)={card}_{Q,{\mathcal {R}}}(]0,p[)}
car
∀
i
∈
N
k
∗
,
v
o
l
1
(
]
(
i
−
1
)
p
,
i
p
[
)
=
v
o
l
1
(
]
0
,
p
[
)
{\displaystyle \forall i\in \mathbb {N} _{k}^{*},\,\,vol^{1}(](i-1)p,ip[)=vol^{1}(]0,p[)}
∀
i
∈
N
k
−
1
∗
,
c
a
r
d
Q
,
R
(
{
i
p
}
)
=
c
a
r
d
Q
,
R
(
{
p
}
)
{\displaystyle \forall i\in \mathbb {N} _{k-1}^{*},\,\,{card}_{Q,{\mathcal {R}}}(\{ip\})={card}_{Q,{\mathcal {R}}}(\{p\})}
c
a
r
d
Q
,
R
(
]
0
,
k
p
[
)
=
∑
i
∈
N
k
∗
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
∑
i
∈
N
k
−
1
∗
c
a
r
d
Q
,
R
(
{
p
}
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(]0,kp[)=\sum _{i\in \mathbb {N} _{k}^{*}}{card}_{Q,{\mathcal {R}}}(]0,p[)+\sum _{i\in \mathbb {N} _{k-1}^{*}}{card}_{Q,{\mathcal {R}}}(\{p\})}}
donc
c
a
r
d
Q
,
R
(
]
0
,
k
p
[
)
=
k
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
(
k
−
1
)
c
a
r
d
Q
,
R
(
{
p
}
)
=
k
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
(
k
−
1
)
=
k
(
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
1
)
−
1
{\displaystyle {card}_{Q,{\mathcal {R}}}(]0,kp[)=k\,\,{card}_{Q,{\mathcal {R}}}(]0,p[)+(k-1)\,\,{card}_{Q,{\mathcal {R}}}(\{p\})=k\,\,{card}_{Q,{\mathcal {R}}}(]0,p[)+(k-1)=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[)+1{\Big )}-1}
•[Point où se rejoignent (1) et (2)]
c
a
r
d
Q
,
R
(
]
0
,
k
p
[
)
+
1
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
1
=
k
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(]0,kp[)+1}{{card}_{Q,{\cal {R}}}(]0,p[)+1}}=k}}
donc
c
a
r
d
Q
,
R
(
]
0
,
k
p
[
)
+
1
c
a
r
d
Q
,
R
(
]
0
,
p
[
)
+
1
=
v
o
l
1
(
]
0
,
k
p
[
)
v
o
l
1
(
]
0
,
p
[
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(]0,kp[)+1}{{card}_{Q,{\cal {R}}}(]0,p[)+1}}={\frac {{vol}^{1}(]0,kp[)}{{vol}^{1}(]0,p[)}}}}
Remarque : On montre facilement le résultat pour
s
∈
Q
{\displaystyle s\in \mathbb {Q} }
s
∈
R
{\displaystyle s\in \mathbb {R} }
v
o
l
1
(
I
)
=
v
o
l
1
(
K
0
,
I
∘
)
{\displaystyle \displaystyle {{vol}^{1}(I)={vol}^{1}(K_{0,{\stackrel {\circ }{I}}})}}
v
o
l
(
J
)
=
v
o
l
1
(
K
0
,
J
∘
)
{\displaystyle \displaystyle {{vol}(J)={vol}^{1}(K_{0,{\stackrel {\circ }{J}}})}}
donc
c
a
r
d
Q
,
R
(
I
∘
)
=
c
a
r
d
Q
,
R
(
K
0
,
I
∘
∘
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})={card}_{Q,{\cal {R}}}({\stackrel {\circ }{K_{0,{\stackrel {\circ }{I}}}}})}}
c
a
r
d
Q
,
R
(
J
∘
)
=
c
a
r
d
Q
,
R
(
K
0
,
J
∘
∘
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})={card}_{Q,{\cal {R}}}({\stackrel {\circ }{K_{0,{\stackrel {\circ }{J}}}}})}}
or
K
0
,
I
∘
∘
=
K
0
,
I
∘
{\displaystyle {\stackrel {\circ }{K_{0,{\stackrel {\circ }{I}}}}}=K_{0,{\stackrel {\circ }{I}}}}
K
0
,
J
∘
∘
=
K
0
,
J
∘
{\displaystyle {\stackrel {\circ }{K_{0,{\stackrel {\circ }{J}}}}}=K_{0,{\stackrel {\circ }{J}}}}
donc
c
a
r
d
Q
,
R
(
I
∘
)
=
c
a
r
d
Q
,
R
(
K
0
,
I
∘
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})={card}_{Q,{\cal {R}}}(K_{0,{\stackrel {\circ }{I}}})}}
c
a
r
d
Q
,
R
(
J
∘
)
=
c
a
r
d
Q
,
R
(
K
0
,
J
∘
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})={card}_{Q,{\cal {R}}}(K_{0,{\stackrel {\circ }{J}}})}}
or
c
a
r
d
Q
(
K
0
,
I
∘
)
+
1
c
a
r
d
Q
(
K
0
,
J
∘
)
+
1
=
v
o
l
1
(
K
0
,
I
∘
)
v
o
l
1
(
K
0
,
J
∘
)
{\displaystyle \displaystyle {{\frac {{card}_{Q}(K_{0,{\stackrel {\circ }{I}}})+1}{{card}_{Q}(K_{0,{\stackrel {\circ }{J}}})+1}}={\frac {{vol}^{1}(K_{0,{\stackrel {\circ }{I}}})}{{vol}^{1}(K_{0,{\stackrel {\circ }{J}}})}}}}
donc
c
a
r
d
Q
,
R
(
I
∘
)
+
1
c
a
r
d
Q
,
R
(
J
∘
)
+
1
=
v
o
l
1
(
I
)
v
o
l
1
(
J
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+1}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+1}}={\frac {{vol}^{1}(I)}{{vol}^{1}(J)}}}}
or
c
a
r
d
Q
,
R
(
I
¯
)
−
1
c
a
r
d
Q
,
R
(
J
¯
)
−
1
=
c
a
r
d
Q
,
R
(
I
∘
)
+
1
c
a
r
d
Q
,
R
(
J
∘
)
+
1
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-1}{{card}_{Q,{\cal {R}}}({\overline {J}})-1}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+1}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+1}}}}
donc
c
a
r
d
Q
,
R
(
I
¯
)
−
1
c
a
r
d
Q
,
R
(
J
¯
)
−
1
=
c
a
r
d
Q
,
R
(
I
∘
)
+
1
c
a
r
d
Q
,
R
(
J
∘
)
+
1
=
v
o
l
1
(
I
)
v
o
l
1
(
J
)
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({\overline {I}})-1}{{card}_{Q,{\cal {R}}}({\overline {J}})-1}}={\frac {{card}_{Q,{\cal {R}}}({\stackrel {\circ }{I}})+1}{{card}_{Q,{\cal {R}}}({\stackrel {\circ }{J}})+1}}={\frac {{vol}^{1}(I)}{{vol}^{1}(J)}}}}
Existence et résultats généraux concernant la F-quantité sur
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Les résultats qui suivent sont ceux donnés par Michel COSTE, dans son PDF "La saga du "cardinal"" (version 4), mais de manière plus rigoureuse, plus détaillée, plus précise, plus développée et mieux formalisée (enfin j'ai fait du mieux que j'ai pu) : N'en déplaise au lecteur contemplatif et admiratif du PDF de vulgarisation de Michel COSTE et aveuglé par ce dernier, il n'appréciera pas, nécessairement et aussi bien, ces résultats, sous cette forme, qui est pourtant leur forme véritable. Et si je n'ai pas fourni les démonstrations de beaucoup d'entre elles, c'est parce que Michel COSTE ne les a pas fournies lui-même et n'a pas donné toutes les références nécessaires.
d
{\displaystyle d}
R
N
{\displaystyle \mathbb {R} ^{N}}
R
N
{\displaystyle \mathbb {R} ^{N}}
R
N
{\displaystyle \mathbb {R} ^{N}}
R
N
{\displaystyle \mathbb {R} ^{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
d
∈
N
N
{\displaystyle d\in \mathbb {N} _{N}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soient
N
∈
N
∗
,
d
∈
N
N
{\displaystyle N\in \mathbb {N} ^{*},\,\,d\in \mathbb {N} _{N}}
Soit
A
N
∈
P
(
R
N
)
{\displaystyle A_{N}\in {\cal {P}}(\mathbb {R} ^{N})}
v
o
l
d
(
A
N
)
{\displaystyle {vol}^{d}(A_{N})}
d
{\displaystyle d}
R
N
{\displaystyle \mathbb {R} ^{N}}
A
N
{\displaystyle A_{N}}
et
d
i
m
(
A
N
)
{\displaystyle {dim}(A_{N})}
R
N
{\displaystyle \mathbb {R} ^{N}}
A
N
{\displaystyle A_{N}}
avec la convention :
d
i
m
(
∅
)
=
+
∞
{\displaystyle {dim}(\emptyset )=+\infty }
Fin du théorème
Remarque : dimension de HAUSDORFF sera toujours à valeur entière positive ou infinie positive.
(Cf. https://homeweb.unifr.ch/manolesc/Pub/teaching/Mesure_integration.pdf )
P
o
l
y
t
o
p
e
s
(
R
N
)
{\displaystyle {{\cal {P}}olytopes}(\mathbb {R} ^{N})}
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
P
o
l
y
t
o
p
e
s
(
R
N
)
{\displaystyle {{\cal {P}}olytopes}(\mathbb {R} ^{N})}
=
d
e
´
f
{
P
N
∈
P
(
R
N
)
|
P
N
p
o
l
y
t
o
p
e
d
e
R
N
(
c
-
a
`
-
d
p
o
l
y
e
`
d
r
e
c
o
m
p
a
c
t
,
c
o
n
v
e
x
e
[
c
o
n
n
e
x
e
]
d
e
R
N
)
}
⋃
{
∅
}
{\displaystyle \displaystyle {{\underset {d{\acute {e}}f}{=}}{\Big \{}{P^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{P^{N}}\,\,polytope\,\,de\,\,\mathbb {R} ^{N}\,\,(c{\mbox{-}}{\grave {a}}{\mbox{-}}d\,\,poly{\grave {e}}dre\,\,compact,\,\,convexe\,\,[connexe]\,\,de\,\,\mathbb {R} ^{N}){\Big \}}\bigcup \{\emptyset \}}}
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
=
d
e
´
f
{
A
N
∈
P
(
R
N
)
|
A
N
s
o
u
s
-
v
a
r
i
e
´
t
e
´
c
o
m
p
a
c
t
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
d
e
R
N
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
o
u
s
a
n
s
b
o
r
d
}
{\displaystyle {\underset {d{\acute {e}}f}{=}}{\Big \{}{A^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{A^{N}}\,\,sous{\text{-}}vari{\acute {e}}t{\acute {e}}\,\,compacte,\,\,convexe\,\,(connexe)\,\,de\,\,\mathbb {R} ^{N},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,ou\,\,sans\,\,bord{\Big \}}}
Fin du théorème
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}
P
V
i
(
R
N
)
{\displaystyle {PV}_{i}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}
=
d
e
´
f
{
P
i
N
∈
P
(
R
N
)
|
P
i
N
p
o
l
y
t
o
p
e
d
e
R
N
(
c
-
a
`
-
d
p
o
l
y
e
`
d
r
e
c
o
m
p
a
c
t
,
c
o
n
v
e
x
e
[
c
o
n
n
e
x
e
]
d
e
R
N
)
e
t
d
i
m
(
P
i
N
)
=
i
}
⋃
{
∅
}
{\displaystyle \displaystyle {{\underset {d{\acute {e}}f}{=}}{\Big \{}{P_{i}^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{P_{i}^{N}}\,\,polytope\,\,de\,\,\mathbb {R} ^{N}\,\,(c{\mbox{-}}{\grave {a}}{\mbox{-}}d\,\,poly{\grave {e}}dre\,\,compact,\,\,convexe\,\,[connexe]\,\,de\,\,\mathbb {R} ^{N})\,\,et\,\,{dim}({P_{i}^{N}})=i{\Big \}}\bigcup \{\emptyset \}}}
=
{
P
i
N
∈
P
o
l
y
t
o
p
e
s
(
R
N
)
|
d
i
m
(
P
i
N
)
=
i
}
⋃
{
∅
}
{\displaystyle \displaystyle {={\Big \{}{P_{i}^{N}}\in {{\cal {P}}olytopes}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{dim}({P_{i}^{N}})=i{\Big \}}\bigcup \{\emptyset \}}}
P
V
i
(
R
N
)
{\displaystyle {PV}_{i}(\mathbb {R} ^{N})}
=
d
e
´
f
{
A
i
N
∈
P
(
R
N
)
|
A
i
N
s
o
u
s
-
v
a
r
i
e
´
t
e
´
c
o
m
p
a
c
t
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
d
e
R
N
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
{\displaystyle \displaystyle {{\underset {d{\acute {e}}f}{=}}{\Big \{}{A_{i}^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{A_{i}^{N}}\,\,sous{\text{-}}vari{\acute {e}}t{\acute {e}}\,\,compacte,\,\,convexe\,\,(connexe)\,\,de\,\,\mathbb {R} ^{N},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)}}
e
t
d
i
m
(
A
i
N
)
=
i
}
⋃
{
∅
}
{\displaystyle \displaystyle {et\,\,{dim}({A_{i}^{N}})=i{\Big \}}\bigcup \{\emptyset \}}}
=
{
A
i
N
∈
P
V
(
R
N
)
|
d
i
m
(
A
i
N
)
=
i
}
⋃
{
∅
}
{\displaystyle \displaystyle {={\Big \{}{A_{i}^{N}}\in {PV}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{dim}({A_{i}^{N}})=i{\Big \}}\bigcup \{\emptyset \}}}
Fin du théorème
P
N
{\displaystyle P_{N}}
L
i
,
N
(
P
N
)
{\displaystyle {\cal {L}}_{i,N}(P_{N})}
P
N
{\displaystyle P_{N}}
P
N
=
P
N
N
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle P_{N}=P_{N}^{N}\in {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
P
N
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle P_{N}\in {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}
∀
r
∈
R
+
,
B
R
N
(
P
N
,
r
)
¯
=
{
x
∈
R
N
|
d
(
P
N
,
x
)
≤
r
}
=
P
N
+
r
B
R
N
(
O
N
,
1
)
¯
{\displaystyle \forall r\in \mathbb {R} _{+},\,\,{\overline {B_{\mathbb {R} ^{N}}(P_{N},r)}}=\{x\in \mathbb {R} ^{N}|d(P_{N},x)\leq r\}=P_{N}+r\,\,{\overline {B_{\mathbb {R} ^{N}}(O_{N},1)}}}
∃
!
(
L
i
,
N
(
P
N
)
)
i
∈
N
N
⊂
R
+
,
∀
r
∈
R
+
,
v
o
l
N
(
B
R
N
(
P
N
,
r
)
¯
)
=
v
o
l
N
(
P
N
+
r
B
R
N
(
O
N
,
1
)
¯
)
=
∑
i
∈
N
N
L
i
,
N
(
P
N
)
r
i
{\displaystyle \displaystyle {\exists !{{\Big (}{\cal {L}}_{i,N}(P_{N}){\Big )}}_{i\in \mathbb {N} _{N}}\subset \mathbb {R} _{+},\,\,\forall r\in \mathbb {R} _{+},\,\,{vol}^{N}{\Big (}{\overline {B_{\mathbb {R} ^{N}}(P_{N},r)}}{\Big )}={vol}^{N}{\Big (}P_{N}+r\,\,{\overline {B_{\mathbb {R} ^{N}}(O_{N},1)}}{\Big )}=\sum _{i\in \mathbb {N} _{N}}{\cal {L}}_{i,N}(P_{N})\,\,r^{i}}}
où
O
N
{\displaystyle O_{N}}
R
N
{\displaystyle {\cal {R}}_{N}}
R
N
{\displaystyle \mathbb {R} ^{N}}
L
0
,
N
(
P
N
)
=
v
o
l
N
(
P
N
)
{\displaystyle {\cal {L}}_{0,N}(P_{N})={vol}^{N}(P_{N})}
L
1
,
N
(
P
N
)
=
v
o
l
N
−
1
(
∂
P
N
)
{\displaystyle {\cal {L}}_{1,N}(P_{N})={vol}^{N-1}(\partial P_{N})}
L
N
,
N
(
P
N
)
=
v
o
l
N
(
B
R
N
(
O
N
,
1
)
¯
)
{\displaystyle {\cal {L}}_{N,N}(P_{N})={vol}^{N}{\Big (}{\overline {B_{\mathbb {R} ^{N}}(O_{N},1)}}{\Big )}}
(
L
i
,
N
(
P
N
)
)
i
∈
N
N
{\displaystyle {{\Big (}{\cal {L}}_{i,N}(P_{N}){\Big )}}_{i\in \mathbb {N} _{N}}}
P
N
{\displaystyle P_{N}}
Fin du théorème
Remarque :
P
i
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle P_{i}^{N}\in {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
La saga du "cardinal" version 4 (voir supra
Remarque :
R
N
{\displaystyle \mathbb {R} ^{N}}
Début d’un théorème
Fin du théorème
L
j
,
i
(
P
i
N
)
,
c
j
,
i
(
P
i
N
)
{\displaystyle {\mathcal {L}}_{j,i}(P_{i}^{N}),\,\,c_{j,i}(P_{i}^{N})}
L
j
,
i
,
c
j
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
R
)
{\displaystyle {\mathcal {L}}_{j,i},\,\,c_{j,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}
P
i
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle P_{i}^{N}\in {{\mathcal {P}}olytopes}_{i}(\mathbb {R} ^{N})}
j
∈
N
i
{\displaystyle j\in \mathbb {N} _{i}}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
L
i
,
N
(
P
N
)
,
c
i
,
N
(
P
N
)
{\displaystyle {\mathcal {L}}_{i,N}(P_{N}),\,\,c_{i,N}(P_{N})}
L
i
,
N
,
c
i
,
N
∈
C
0
(
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
R
)
{\displaystyle {\mathcal {L}}_{i,N},\,\,c_{i,N}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}
P
N
=
P
N
N
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle P_{N}=P_{N}^{N}\in {{\mathcal {P}}olytopes}_{N}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
P
N
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle \displaystyle {P_{N}\in {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}}
∀
i
∈
N
N
,
c
i
,
N
(
P
N
)
=
L
N
−
i
,
N
(
P
N
)
β
(
N
−
i
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,c_{i,N}(P_{N})={\frac {{\mathcal {L}}_{N-i,N}(P_{N})}{\beta (N-i)}}}}
où
∀
i
∈
N
N
,
β
(
i
)
=
v
o
l
i
(
B
R
i
(
O
i
,
1
)
¯
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,\beta (i)={vol}^{i}{\Big (}{\overline {B_{\mathbb {R} ^{i}}(O_{i},1)}}{\Big )}}}
∀
i
∈
N
N
,
O
i
{\displaystyle \forall i\in \mathbb {N} _{N},\,\,O_{i}}
R
i
{\displaystyle {\cal {R}}_{i}}
R
i
{\displaystyle \mathbb {R} ^{i}}
(
L
i
,
N
(
P
N
)
)
i
∈
N
N
{\displaystyle {{\Big (}{\mathcal {L}}_{i,N}(P_{N}){\Big )}}_{i\in \mathbb {N} _{N}}}
P
N
{\displaystyle P_{N}}
L
0
,
N
(
P
N
)
=
v
o
l
N
(
P
N
)
{\displaystyle {\cal {L}}_{0,N}(P_{N})={vol}^{N}(P_{N})}
L
1
,
N
(
P
N
)
=
v
o
l
N
−
1
(
∂
P
N
)
{\displaystyle {\cal {L}}_{1,N}(P_{N})={vol}^{N-1}(\partial P_{N})}
L
N
,
N
(
P
N
)
=
v
o
l
N
(
B
R
N
(
O
N
,
1
)
¯
)
{\displaystyle {\cal {L}}_{N,N}(P_{N})={vol}^{N}{\Big (}{\overline {B_{\mathbb {R} ^{N}}(O_{N},1)}}{\Big )}}
et on a :
c
0
,
N
(
P
N
)
=
1
{\displaystyle c_{0,N}(P_{N})=1}
∀
i
∈
N
N
,
L
i
,
N
:
P
o
l
y
t
o
p
e
s
N
(
R
N
)
⟶
R
:
P
N
⟼
L
i
,
N
(
P
N
)
{\displaystyle \forall i\in \mathbb {N} _{N},\,\,{\cal {L}}_{i,N}\,\,:\,\,{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,\mathbb {R} \,\,:\,\,P_{N}\,\,\longmapsto \,\,{\cal {L}}_{i,N}(P_{N})}
∀
i
∈
N
N
,
c
i
,
N
:
P
o
l
y
t
o
p
e
s
N
(
R
N
)
⟶
R
:
P
N
⟼
c
i
,
N
(
P
N
)
{\displaystyle \forall i\in \mathbb {N} _{N},\,\,c_{i,N}\,\,:\,\,{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,\mathbb {R} \,\,:\,\,P_{N}\,\,\longmapsto \,\,c_{i,N}(P_{N})}
∀
i
∈
N
N
,
L
i
,
N
∈
C
0
(
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
R
)
{\displaystyle {\forall i\in \mathbb {N} _{N},\,\,{\cal {L}}_{i,N}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
∀
i
∈
N
N
,
c
i
,
N
∈
C
0
(
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
R
)
{\displaystyle {\forall i\in \mathbb {N} _{N},\,\,c_{i,N}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
2) Soit
P
i
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle \displaystyle {P_{i}^{N}\in {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}
∀
j
∈
N
i
,
c
j
,
i
(
P
i
N
)
=
L
i
−
j
,
i
(
P
i
N
)
β
(
i
−
j
)
{\displaystyle \displaystyle {\forall j\in \mathbb {N} _{i},\,\,c_{j,i}(P_{i}^{N})={\frac {{\mathcal {L}}_{i-j,i}(P_{i}^{N})}{\beta (i-j)}}}}
où
∀
j
∈
N
i
,
β
(
j
)
=
v
o
l
j
(
B
R
j
(
O
j
,
1
)
¯
)
{\displaystyle \displaystyle {\forall j\in \mathbb {N} _{i},\,\,\beta (j)={vol}^{j}{\Big (}{\overline {B_{\mathbb {R} ^{j}}(O_{j},1)}}{\Big )}}}
∀
j
∈
N
i
,
O
j
{\displaystyle \forall j\in \mathbb {N} _{i},\,\,O_{j}}
R
j
{\displaystyle {\cal {R}}_{j}}
R
j
{\displaystyle \mathbb {R} ^{j}}
(
L
j
,
i
(
P
i
N
)
)
j
∈
N
i
{\displaystyle {{\Big (}{\mathcal {L}}_{j,i}(P_{i}^{N}){\Big )}}_{j\in \mathbb {N} _{i}}}
P
i
N
{\displaystyle P_{i}^{N}}
L
0
,
i
(
P
i
N
)
=
v
o
l
i
(
P
i
N
)
{\displaystyle {\cal {L}}_{0,i}(P_{i}^{N})={vol}^{i}(P_{i}^{N})}
L
1
,
i
(
P
i
N
)
=
v
o
l
i
−
1
(
∂
P
i
N
)
{\displaystyle {\cal {L}}_{1,i}(P_{i}^{N})={vol}^{i-1}(\partial P_{i}^{N})}
L
i
,
i
(
P
i
N
)
=
v
o
l
i
(
B
R
i
(
O
i
,
1
)
¯
)
{\displaystyle {\cal {L}}_{i,i}(P_{i}^{N})={vol}^{i}{\Big (}{\overline {B_{\mathbb {R} ^{i}}(O_{i},1)}}{\Big )}}
et on a :
c
0
,
i
(
P
i
N
)
=
1
{\displaystyle c_{0,i}(P_{i}^{N})=1}
∀
j
∈
N
i
,
L
j
,
i
:
P
o
l
y
t
o
p
e
s
i
(
R
N
)
⟶
R
:
P
i
N
⟼
L
j
,
i
(
P
i
N
)
{\displaystyle \forall j\in \mathbb {N} _{i},\,\,{\cal {L}}_{j,i}\,\,:\,\,{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,\mathbb {R} \,\,:\,\,P_{i}^{N}\,\,\longmapsto \,\,{\cal {L}}_{j,i}(P_{i}^{N})}
et où
∀
j
∈
N
i
,
c
j
,
i
:
P
o
l
y
t
o
p
e
s
i
(
R
N
)
⟶
R
:
P
i
N
⟼
c
j
,
i
(
P
i
N
)
{\displaystyle \forall j\in \mathbb {N} _{i},\,\,c_{j,i}\,\,:\,\,{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,\mathbb {R} \,\,:\,\,P_{i}^{N}\,\,\longmapsto \,\,c_{j,i}(P_{i}^{N})}
∀
j
∈
N
i
,
L
j
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
R
)
{\displaystyle {\forall j\in \mathbb {N} _{i},\,\,{\cal {L}}_{j,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
∀
j
∈
N
i
,
c
j
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
R
)
{\displaystyle {\forall j\in \mathbb {N} _{i},\,\,c_{j,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
Fin du théorème
Démonstration : Théorème de HADWIGER (en anglais) .
Remarque :
c
a
r
d
Q
∈
C
0
(
P
o
l
y
t
o
p
e
s
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{card}_{Q}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}(\mathbb {R} ^{N}),F{\Big )}}}
c
a
r
d
Q
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{card}_{Q,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),F{\Big )}}}
P
i
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle P_{i}^{N}\in {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
P
N
=
P
N
N
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle P_{N}=P_{N}^{N}\in {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[
0
,
1
[
{\displaystyle [0,1[}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
∃
!
c
a
r
d
Q
,
N
:
P
o
l
y
t
o
p
e
s
N
(
R
N
)
⟶
F
,
{\displaystyle \exists !{card}_{Q,N}\,\,:\,\,{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,F,}
telle que
∀
R
N
′
r
e
p
e
`
r
e
o
r
t
h
o
n
o
r
m
e
´
d
e
R
N
,
c
a
r
d
Q
,
R
N
′
|
P
o
l
y
t
o
p
e
s
N
(
R
N
)
=
c
a
r
d
Q
,
N
{\displaystyle \forall {{\cal {R}}_{N}}'\,\,rep{\grave {e}}re\,\,orthonorm{\acute {e}}\,\,de\,\,\mathbb {R} ^{N},\,\,{{card}_{Q,{{\cal {R}}_{N}}'}}_{{\Big |}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}={card}_{Q,N}}
et telle que
∀
P
N
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
c
a
r
d
Q
,
N
(
P
N
)
=
∑
i
∈
N
N
c
i
,
N
(
P
N
)
c
a
r
d
Q
,
1
i
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall P_{N}\in {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),\,\,{card}_{Q,N}(P_{N})=\sum _{i\in \mathbb {N} _{N}}c_{i,N}(P_{N})\,\,{card}_{Q,1}^{i}([0,1[)}}
c
a
r
d
Q
,
N
∈
C
0
(
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{card}_{Q,N}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),F{\Big )}}}
2)
∃
!
c
a
r
d
Q
,
i
:
P
o
l
y
t
o
p
e
s
i
(
R
N
)
⟶
F
,
{\displaystyle \exists !{card}_{Q,i}\,\,:\,\,{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,F,}
telle que
∀
R
N
′
r
e
p
e
`
r
e
o
r
t
h
o
n
o
r
m
e
´
d
e
R
N
,
c
a
r
d
Q
,
R
N
′
|
P
o
l
y
t
o
p
e
s
i
(
R
N
)
=
c
a
r
d
Q
,
i
{\displaystyle \forall {{\cal {R}}_{N}}'\,\,rep{\grave {e}}re\,\,orthonorm{\acute {e}}\,\,de\,\,\mathbb {R} ^{N},\,\,{{card}_{Q,{{\cal {R}}_{N}}'}}_{{\Big |}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}={card}_{Q,i}}
et telle que
∀
P
i
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
c
a
r
d
Q
,
N
(
P
i
N
)
=
∑
j
∈
N
i
c
j
,
i
(
P
i
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall P_{i}^{N}\in {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\,\,{card}_{Q,N}(P_{i}^{N})=\sum _{j\in \mathbb {N} _{i}}c_{j,i}(P_{i}^{N})\,\,{card}_{Q,1}^{j}([0,1[)}}
c
a
r
d
Q
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{card}_{Q,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),F{\Big )}}}
Remarque : On peut aussi poser
c
a
r
d
Q
:
P
o
l
y
t
o
p
e
s
(
R
N
)
=
⨆
i
∈
N
N
P
o
l
y
t
o
p
e
s
i
(
R
N
)
⟶
F
{\displaystyle \displaystyle {{card}_{Q}\,\,:\,\,{{\cal {P}}olytopes}(\mathbb {R} ^{N})=\bigsqcup _{i\in \mathbb {N} _{N}}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})\longrightarrow F}}
∀
i
∈
N
N
,
c
a
r
d
Q
|
P
o
l
y
t
o
p
e
s
i
(
R
N
)
=
c
a
r
d
Q
,
i
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,{{card}_{Q}}_{{\Big |}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}={card}_{Q,i}}}
et telle que
∀
i
∈
N
N
,
∀
P
i
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
c
a
r
d
Q
,
i
(
P
i
N
)
=
∑
j
∈
N
i
c
j
,
i
(
P
i
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,\forall P_{i}^{N}\in {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\,\,{card}_{Q,i}(P_{i}^{N})=\sum _{j\in \mathbb {N} _{i}}c_{j,i}(P_{i}^{N})\,\,{card}_{Q,1}^{j}([0,1[)}}
c
a
r
d
Q
∈
C
0
(
P
o
l
y
t
o
p
e
s
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{card}_{Q}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}(\mathbb {R} ^{N}),F{\Big )}}}
Fin du théorème
Démonstration : Théorème de HADWIGER (en anglais) et le lemme précédent :
Cf. La saga du "cardinal" version 4, Théorème de HADWIGER (voir supra
Remarque :
Remarque :
F
{\displaystyle F}
N
⨆
+
∞
{\displaystyle \mathbb {N} \bigsqcup +\infty }
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle +\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
Remarque :
∀
i
∈
N
N
,
c
i
,
N
(
P
N
)
=
L
i
,
N
(
P
N
)
β
(
i
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,c_{i,N}(P_{N})={\frac {{\mathcal {L}}_{i,N}(P_{N})}{\beta (i)}}}}
P
o
l
y
t
o
p
e
s
i
(
R
N
)
¯
P
V
i
(
R
N
)
=
P
V
i
(
R
N
)
{\displaystyle \displaystyle {{\overline {{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}^{{PV}_{i}(\mathbb {R} ^{N})}={PV}_{i}(\mathbb {R} ^{N})}}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
P
o
l
y
t
o
p
e
s
N
(
R
N
)
¯
P
V
N
(
R
N
)
=
P
V
N
(
R
N
)
{\displaystyle \displaystyle {{\overline {{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}}^{{PV}_{N}(\mathbb {R} ^{N})}={PV}_{N}(\mathbb {R} ^{N})}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
P
o
l
y
t
o
p
e
s
N
(
R
N
)
¯
P
V
N
(
R
N
)
=
P
V
N
(
R
N
)
{\displaystyle \displaystyle {{\overline {{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}}^{{PV}_{N}(\mathbb {R} ^{N})}={PV}_{N}(\mathbb {R} ^{N})}}
∀
A
N
∈
P
V
N
(
R
N
)
{\displaystyle \displaystyle {\forall A_{N}\in {PV}_{N}(\mathbb {R} ^{N})}}
∃
(
P
N
,
n
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
A
N
=
lim
n
→
+
∞
P
N
,
n
{\displaystyle \displaystyle {\exists {(P_{N,n})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),\,\,A_{N}=\lim _{n\rightarrow +\infty }P_{N,n}}}
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle \displaystyle {{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}}
P
V
N
(
R
N
)
{\displaystyle {PV}_{N}(\mathbb {R} ^{N})}
P
o
l
y
t
o
p
e
s
i
(
R
N
)
¯
P
V
i
(
R
N
)
=
P
V
i
(
R
N
)
{\displaystyle \displaystyle {{\overline {{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}^{{PV}_{i}(\mathbb {R} ^{N})}={PV}_{i}(\mathbb {R} ^{N})}}
∀
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle \displaystyle {\forall A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}}
∃
(
P
i
,
n
N
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
A
i
N
=
lim
n
→
+
∞
P
i
,
n
N
{\displaystyle \displaystyle {\exists {(P_{i,n}^{N})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\,\,A_{i}^{N}=\lim _{n\rightarrow +\infty }P_{i,n}^{N}}}
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle \displaystyle {{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}
P
V
i
(
R
N
)
{\displaystyle {PV}_{i}(\mathbb {R} ^{N})}
Fin du théorème
voir supra
L
j
,
i
~
(
A
i
N
)
,
c
j
,
i
~
(
A
i
N
)
{\displaystyle {\widetilde {{\mathcal {L}}_{j,i}}}(A_{i}^{N}),\,\,{\widetilde {c_{j,i}}}(A_{i}^{N})}
L
j
,
i
~
,
c
j
,
i
~
∈
C
0
(
P
V
i
(
R
N
)
,
R
)
{\displaystyle {\widetilde {{\mathcal {L}}_{j,i}}},\,\,{\widetilde {c_{j,i}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}
j
∈
N
i
{\displaystyle j\in \mathbb {N} _{i}}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
L
i
,
N
~
(
A
N
)
,
c
i
,
N
~
(
A
N
)
{\displaystyle {\widetilde {{\mathcal {L}}_{i,N}}}(A_{N}),\,\,{\widetilde {c_{i,N}}}(A_{N})}
L
i
,
N
~
,
c
i
,
N
~
∈
C
0
(
P
V
N
(
R
N
)
,
R
)
{\displaystyle {\widetilde {{\mathcal {L}}_{i,N}}},\,\,{\widetilde {c_{i,N}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}
A
N
=
A
N
N
∈
P
V
N
(
R
N
)
{\displaystyle A_{N}=A_{N}^{N}\in {PV}_{N}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
Soit
A
N
∈
P
V
N
(
R
N
)
{\displaystyle \displaystyle {A_{N}\in {PV}_{N}(\mathbb {R} ^{N})}}
∃
(
P
N
,
n
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle \displaystyle {\exists {(P_{N,n})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}}
A
N
=
lim
n
→
+
∞
P
N
,
n
{\displaystyle \displaystyle {A_{N}=\lim _{n\rightarrow +\infty }P_{N,n}}}
(*1-1)
∀
i
∈
N
N
,
L
i
,
N
∈
C
0
(
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
R
)
{\displaystyle {\forall i\in \mathbb {N} _{N},\,\,{\cal {L}}_{i,N}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
(
L
i
,
N
~
(
A
N
)
)
i
∈
N
N
⊂
R
{\displaystyle {{\Big (}{\widetilde {{\cal {L}}_{i,N}}}(A_{N}){\Big )}}_{i\in \mathbb {N} _{N}}\subset \mathbb {R} }
∀
i
∈
N
N
,
L
i
,
N
~
(
A
N
)
=
L
i
,
N
~
(
lim
n
→
+
∞
P
N
,
n
)
=
lim
n
→
+
∞
L
i
,
N
(
P
N
,
n
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,{\widetilde {{\cal {L}}_{i,N}}}(A_{N})={\widetilde {{\cal {L}}_{i,N}}}(\lim _{n\rightarrow +\infty }P_{N,n})=\displaystyle {\lim _{n\rightarrow +\infty }{\cal {L}}_{i,N}(P_{N,n})}}}
et montrer que cette définition ne dépend pas de la suite
(
P
N
,
n
)
n
∈
N
{\displaystyle {(P_{N,n})}_{n\in \mathbb {N} }}
(*2-1)
∀
i
∈
N
N
,
c
i
,
N
∈
C
0
(
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
R
)
{\displaystyle {\forall i\in \mathbb {N} _{N},\,\,c_{i,N}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
(
c
i
,
N
~
(
A
N
)
)
i
∈
N
N
⊂
R
{\displaystyle {{\Big (}{\widetilde {c_{i,N}}}(A_{N}){\Big )}}_{i\in \mathbb {N} _{N}}\subset \mathbb {R} }
∀
i
∈
N
N
,
c
i
,
N
~
(
A
N
)
=
c
i
,
N
~
(
lim
n
→
+
∞
P
N
,
n
)
=
lim
n
→
+
∞
c
i
,
N
(
P
N
,
n
)
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,{\widetilde {c_{i,N}}}(A_{N})={\widetilde {c_{i,N}}}(\lim _{n\rightarrow +\infty }P_{N,n})=\displaystyle {\lim _{n\rightarrow +\infty }c_{i,N}(P_{N,n})}}}
et montrer que cette définition ne dépend pas de la suite
(
P
N
,
n
)
n
∈
N
{\displaystyle {(P_{N,n})}_{n\in \mathbb {N} }}
∀
i
∈
N
N
,
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},}}
∃
!
L
i
,
N
~
∈
C
0
(
P
V
N
(
R
N
)
,
R
)
{\displaystyle \displaystyle {\exists !{\widetilde {{\cal {L}}_{i,N}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
telle que
L
i
,
N
~
|
P
o
l
y
t
o
p
e
s
N
(
R
N
)
=
L
i
,
N
{\displaystyle \displaystyle {{\widetilde {{\cal {L}}_{i,N}}}_{|{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}={\cal {L}}_{i,N}}}
L
i
,
N
~
:
P
V
N
(
R
N
)
⟶
R
:
A
N
⟼
L
i
,
N
~
(
A
N
)
{\displaystyle \displaystyle {{\widetilde {{\cal {L}}_{i,N}}}\,\,:\,\,{PV}_{N}(\mathbb {R} ^{N})\,\,\longrightarrow \mathbb {R} \,\,:\,\,A_{N}\,\,\longmapsto \,\,{\widetilde {{\cal {L}}_{i,N}}}(A_{N})}}
L
i
,
N
~
(
A
N
)
{\displaystyle {\widetilde {{\cal {L}}_{i,N}}}(A_{N})}
∀
i
∈
N
N
,
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},}}
∃
!
c
i
,
N
~
∈
C
0
(
P
V
N
(
R
N
)
,
R
)
{\displaystyle \displaystyle {\exists !{\widetilde {c_{i,N}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{N}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
telle que
c
i
,
N
~
|
P
o
l
y
t
o
p
e
s
N
(
R
N
)
=
c
i
,
N
{\displaystyle \displaystyle {{\widetilde {c_{i,N}}}_{|{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}=c_{i,N}}}
c
i
,
N
~
:
P
V
N
(
R
N
)
⟶
R
:
A
N
⟼
c
i
,
N
~
(
A
N
)
{\displaystyle \displaystyle {{\widetilde {c_{i,N}}}\,\,:\,\,{PV}_{N}(\mathbb {R} ^{N})\,\,\longrightarrow \mathbb {R} \,\,:\,\,A_{N}\,\,\longmapsto \,\,{\widetilde {c_{i,N}}}(A_{N})}}
c
i
,
N
~
(
A
N
)
{\displaystyle {\widetilde {c_{i,N}}}(A_{N})}
et on a :
L
0
,
N
~
(
A
N
)
=
v
o
l
N
(
A
N
)
{\displaystyle {\widetilde {{\cal {L}}_{0,N}}}(A_{N})={vol}^{N}(A_{N})}
L
1
,
N
~
(
A
N
)
=
v
o
l
N
−
1
(
∂
A
N
)
{\displaystyle {\widetilde {{\cal {L}}_{1,N}}}(A_{N})={vol}^{N-1}(\partial A_{N})}
L
N
,
N
~
(
A
N
)
=
v
o
l
N
(
B
R
N
(
O
N
,
1
)
¯
)
{\displaystyle {\widetilde {{\cal {L}}_{N,N}}}(A_{N})={vol}^{N}{\Big (}{\overline {B_{\mathbb {R} ^{N}}(O_{N},1)}}{\Big )}}
et on a :
c
0
,
N
~
(
A
N
)
=
1
{\displaystyle {\widetilde {c_{0,N}}}(A_{N})=1}
2) D'après la proposition précédente :
Soit
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle \displaystyle {A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}}
∃
(
P
i
,
n
N
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
i
(
R
N
)
{\displaystyle \displaystyle {\exists {(P_{i,n}^{N})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytope}_{i}(\mathbb {R} ^{N})}}
A
i
N
=
lim
n
→
+
∞
P
i
,
n
N
{\displaystyle \displaystyle {A_{i}^{N}=\lim _{n\rightarrow +\infty }P_{i,n}^{N}}}
(*1-2)
∀
j
∈
N
i
,
L
j
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
R
)
{\displaystyle {\forall j\in \mathbb {N} _{i},\,\,{\cal {L}}_{j,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
(
L
j
,
i
~
(
A
i
N
)
)
j
∈
N
i
⊂
R
{\displaystyle {{\Big (}{\widetilde {{\cal {L}}_{j,i}}}(A_{i}^{N}){\Big )}}_{j\in \mathbb {N} _{i}}\subset \mathbb {R} }
∀
j
∈
N
i
,
L
j
,
i
~
(
A
i
N
)
=
L
j
,
i
~
(
lim
n
→
+
∞
P
i
,
n
N
)
=
lim
n
→
+
∞
L
j
,
i
(
P
i
,
n
N
)
{\displaystyle \displaystyle {\forall j\in \mathbb {N} _{i},\,\,{\widetilde {{\cal {L}}_{j,i}}}(A_{i}^{N})={\widetilde {{\cal {L}}_{j,i}}}(\lim _{n\rightarrow +\infty }P_{i,n}^{N})=\displaystyle {\lim _{n\rightarrow +\infty }{\cal {L}}_{j,i}(P_{i,n}^{N})}}}
et montrer que cette définition ne dépend pas de la suite
(
P
i
,
n
N
)
n
∈
N
{\displaystyle {(P_{i,n}^{N})}_{n\in \mathbb {N} }}
(*2-2)
∀
j
∈
N
i
,
c
j
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
R
)
{\displaystyle {\forall j\in \mathbb {N} _{i},\,\,c_{j,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
(
c
j
,
i
~
(
A
i
N
)
)
j
∈
N
i
⊂
R
{\displaystyle {{\Big (}{\widetilde {c_{j,i}}}(A_{i}^{N}){\Big )}}_{j\in \mathbb {N} _{i}}\subset \mathbb {R} }
∀
j
∈
N
i
,
c
j
,
i
~
(
A
i
N
)
=
c
j
,
i
~
(
lim
n
→
+
∞
P
i
,
n
N
)
=
lim
n
→
+
∞
c
j
,
i
(
P
i
,
n
N
)
{\displaystyle \displaystyle {\forall j\in \mathbb {N} _{i},\,\,{\widetilde {c_{j,i}}}(A_{i}^{N})={\widetilde {c_{j,i}}}(\lim _{n\rightarrow +\infty }P_{i,n}^{N})=\displaystyle {\lim _{n\rightarrow +\infty }c_{j,i}(P_{i,n}^{N})}}}
et montrer que cette définition ne dépend pas de la suite
(
P
i
,
n
N
)
n
∈
N
{\displaystyle {(P_{i,n}^{N})}_{n\in \mathbb {N} }}
∀
j
∈
N
i
,
{\displaystyle \displaystyle {\forall j\in \mathbb {N} _{i},}}
∃
!
L
j
,
i
~
∈
C
0
(
P
V
i
(
R
N
)
,
R
)
{\displaystyle \displaystyle {\exists !{\widetilde {{\cal {L}}_{j,i}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
telle que
L
j
,
i
~
|
P
o
l
y
t
o
p
e
s
i
(
R
N
)
=
L
j
,
i
{\displaystyle \displaystyle {{\widetilde {{\cal {L}}_{j,i}}}_{|{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}={\cal {L}}_{j,i}}}
L
j
,
i
~
:
P
V
i
(
R
N
)
⟶
R
:
A
i
N
⟼
L
j
,
i
~
(
A
i
N
)
{\displaystyle \displaystyle {{\widetilde {{\cal {L}}_{j,i}}}\,\,:\,\,{PV}_{i}(\mathbb {R} ^{N})\,\,\longrightarrow \mathbb {R} \,\,:\,\,A_{i}^{N}\,\,\longmapsto \,\,{\widetilde {{\cal {L}}_{j,i}}}(A_{i}^{N})}}
L
j
,
i
~
(
A
i
N
)
{\displaystyle {\widetilde {{\cal {L}}_{j,i}}}(A_{i}^{N})}
∀
j
∈
N
i
,
{\displaystyle \displaystyle {\forall j\in \mathbb {N} _{i},}}
∃
!
c
j
,
i
~
∈
C
0
(
P
V
i
(
R
N
)
,
R
)
{\displaystyle \displaystyle {\exists !{\widetilde {c_{j,i}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{i}(\mathbb {R} ^{N}),\mathbb {R} {\Big )}}}
telle que
c
j
,
i
~
|
P
o
l
y
t
o
p
e
s
i
(
R
N
)
=
c
j
,
i
{\displaystyle \displaystyle {{\widetilde {c_{j,i}}}_{|{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}=c_{j,i}}}
c
j
,
i
~
:
P
V
i
(
R
N
)
⟶
R
:
A
N
⟼
c
j
,
i
~
(
A
i
N
)
{\displaystyle \displaystyle {{\widetilde {c_{j,i}}}\,\,:\,\,{PV}_{i}(\mathbb {R} ^{N})\,\,\longrightarrow \mathbb {R} \,\,:\,\,A_{N}\,\,\longmapsto \,\,{\widetilde {c_{j,i}}}(A_{i}^{N})}}
c
j
,
i
~
(
A
i
N
)
{\displaystyle {\widetilde {c_{j,i}}}(A_{i}^{N})}
et on a :
L
0
,
i
~
(
A
i
N
)
=
v
o
l
i
(
A
i
N
)
{\displaystyle {\widetilde {{\cal {L}}_{0,i}}}(A_{i}^{N})={vol}^{i}(A_{i}^{N})}
L
1
,
i
~
(
A
i
N
)
=
v
o
l
i
−
1
(
∂
A
i
N
)
{\displaystyle {\widetilde {{\cal {L}}_{1,i}}}(A_{i}^{N})={vol}^{i-1}(\partial A_{i}^{N})}
L
i
,
i
~
(
A
i
N
)
=
v
o
l
i
(
B
R
i
(
O
i
,
1
)
¯
)
{\displaystyle {\widetilde {{\cal {L}}_{i,i}}}(A_{i}^{N})={vol}^{i}{\Big (}{\overline {B_{\mathbb {R} ^{i}}(O_{i},1)}}{\Big )}}
et
c
0
,
i
~
(
A
i
N
)
=
1
{\displaystyle {\widetilde {c_{0,i}}}(A_{i}^{N})=1}
Fin du théorème
Remarque :
c
a
r
d
Q
~
∈
C
0
(
P
V
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}\in {\mathcal {C}}^{0}{\Big (}{PV}(\mathbb {R} ^{N}),F{\Big )}}}
c
a
r
d
Q
,
i
~
∈
C
0
(
P
V
i
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,i}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{i}(\mathbb {R} ^{N}),F{\Big )}}}
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
A
N
=
A
N
N
∈
P
V
N
(
R
N
)
{\displaystyle A_{N}=A_{N}^{N}\in {PV}_{N}(\mathbb {R} ^{N})}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[
0
,
1
[
{\displaystyle [0,1[}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
Soit
A
N
∈
P
V
N
(
R
N
)
{\displaystyle \displaystyle {A_{N}\in {PV}_{N}(\mathbb {R} ^{N})}}
∃
(
P
N
,
n
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle \displaystyle {\exists {(P_{N,n})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}}
A
N
=
lim
n
→
+
∞
P
N
,
n
{\displaystyle \displaystyle {A_{N}=\lim _{n\rightarrow +\infty }P_{N,n}}}
(*3-1)
c
a
r
d
Q
,
N
∈
C
0
(
P
o
l
y
t
o
p
e
s
N
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{card}_{Q,N}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N}),F{\Big )}}}
c
a
r
d
Q
,
N
~
(
A
N
)
=
c
a
r
d
Q
,
N
~
(
lim
n
→
+
∞
P
N
,
n
)
=
lim
n
→
+
∞
c
a
r
d
Q
,
N
(
P
N
,
n
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,N}}}(A_{N})={\widetilde {{card}_{Q,N}}}(\lim _{n\rightarrow +\infty }P_{N,n})=\displaystyle {\lim _{n\rightarrow +\infty }{card}_{Q,N}(P_{N,n})}}}
et montrer que cette définition ne dépend pas de la suite
(
P
N
,
n
)
n
∈
N
{\displaystyle {(P_{N,n})}_{n\in \mathbb {N} }}
∀
n
∈
N
,
c
a
r
d
Q
,
N
(
P
N
,
n
)
=
∑
i
∈
N
N
c
i
,
N
(
P
N
,
n
)
c
a
r
d
Q
,
1
i
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall n\in \mathbb {N} ,\,\,{card}_{Q,N}(P_{N,n})=\sum _{i\in \mathbb {N} _{N}}c_{i,N}(P_{N,n})\,\,{card}_{Q,1}^{i}([0,1[)}}
∃
!
c
a
r
d
Q
,
N
~
∈
C
0
(
P
V
N
(
R
N
)
,
F
)
{\displaystyle \exists !{\widetilde {{card}_{Q,N}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{N}(\mathbb {R} ^{N}),F{\Big )}}
telle que
∀
R
N
′
r
e
p
e
`
r
e
o
r
t
h
o
n
o
r
m
e
´
d
e
R
N
,
c
a
r
d
Q
,
R
N
′
|
P
V
N
(
R
N
)
=
c
a
r
d
Q
,
N
~
{\displaystyle \forall {{\cal {R}}_{N}}'\,\,rep{\grave {e}}re\,\,orthonorm{\acute {e}}\,\,de\,\,\mathbb {R} ^{N},\,\,{{card}_{Q,{{\cal {R}}_{N}}'}}_{{\Big |}{PV}_{N}(\mathbb {R} ^{N})}={\widetilde {{card}_{Q,N}}}}
et telle que
c
a
r
d
Q
,
N
~
|
P
o
l
y
t
o
p
e
N
(
R
N
)
=
c
a
r
d
Q
,
N
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,N}}}_{|{{\cal {P}}olytope}_{N}(\mathbb {R} ^{N})}={card}_{Q,N}}}
[comme, on a (*1-1), (*2-1) et (*3-1)]
∀
A
N
∈
P
V
N
(
R
N
)
,
{\displaystyle \displaystyle {\forall A_{N}\in {PV}_{N}(\mathbb {R} ^{N})},}
∃
(
P
N
,
n
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle \displaystyle {\exists {(P_{N,n})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{N}(\mathbb {R} ^{N})}}
A
N
=
lim
n
→
+
∞
P
N
,
n
,
{\displaystyle \displaystyle {A_{N}=\lim _{n\rightarrow +\infty }P_{N,n},}}
c
a
r
d
Q
,
N
~
(
A
N
)
=
c
a
r
d
Q
,
N
~
(
lim
n
→
+
∞
P
N
,
n
)
=
lim
n
→
+
∞
c
a
r
d
Q
,
N
(
P
N
,
n
)
=
lim
n
→
+
∞
∑
i
∈
N
N
c
i
,
N
(
P
N
,
n
)
c
a
r
d
Q
,
1
i
(
[
0
,
1
[
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,N}}}(A_{N})={\widetilde {{card}_{Q,N}}}(\lim _{n\rightarrow +\infty }P_{N,n})=\lim _{n\rightarrow +\infty }{card}_{Q,N}(P_{N,n})=\lim _{n\rightarrow +\infty }\sum _{i\in \mathbb {N} _{N}}c_{i,N}(P_{N,n})\,\,{card}_{Q,1}^{i}([0,1[)}}
=
∑
i
∈
N
N
lim
n
→
+
∞
c
i
,
N
(
P
N
,
n
)
c
a
r
d
Q
,
1
i
(
[
0
,
1
[
)
=
∑
i
∈
N
N
c
i
,
N
~
(
lim
n
→
+
∞
P
N
,
n
)
c
a
r
d
Q
,
1
i
(
[
0
,
1
[
)
=
∑
i
∈
N
N
c
i
,
N
~
(
A
N
)
c
a
r
d
Q
,
1
i
(
[
0
,
1
[
)
=
∑
i
∈
N
N
c
i
,
N
~
(
A
N
)
c
a
r
d
Q
,
1
~
i
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=\sum _{i\in \mathbb {N} _{N}}\lim _{n\rightarrow +\infty }c_{i,N}(P_{N,n})\,\,{card}_{Q,1}^{i}([0,1[)=\sum _{i\in \mathbb {N} _{N}}{\widetilde {c_{i,N}}}(\lim _{n\rightarrow +\infty }P_{N,n})\,\,{card}_{Q,1}^{i}([0,1[)=\sum _{i\in \mathbb {N} _{N}}{\widetilde {c_{i,N}}}(A_{N})\,\,{card}_{Q,1}^{i}([0,1[)=\sum _{i\in \mathbb {N} _{N}}{\widetilde {c_{i,N}}}(A_{N})\,\,{\widetilde {{card}_{Q,1}}}^{i}([0,1[)}}
∀
A
N
∈
P
V
N
(
R
N
)
,
c
a
r
d
Q
,
N
~
(
A
N
)
=
∑
i
∈
N
N
c
i
,
N
~
(
A
N
)
c
a
r
d
Q
,
1
~
i
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall A_{N}\in {PV}_{N}(\mathbb {R} ^{N}),\,\,{\widetilde {{card}_{Q,N}}}(A_{N})=\sum _{i\in \mathbb {N} _{N}}{\widetilde {c_{i,N}}}(A_{N})\,\,{\widetilde {{card}_{Q,1}}}^{i}([0,1[)}}
c
a
r
d
Q
,
N
~
:
P
V
N
(
R
N
)
⟶
F
:
A
N
↦
c
a
r
d
Q
,
N
~
(
A
N
)
{\displaystyle {\widetilde {{card}_{Q,N}}}\,\,:{PV}_{N}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,F\,\,:\,\,A_{N}\,\,\mapsto \,\,{\widetilde {{card}_{Q,N}}}(A_{N})}
c
a
r
d
Q
,
N
~
(
A
N
)
{\displaystyle {\widetilde {{card}_{Q,N}}}(A_{N})}
Soit
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle \displaystyle {A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}}
∃
(
P
i
,
n
N
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle \displaystyle {\exists {(P_{i,n}^{N})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}
A
i
N
=
lim
n
→
+
∞
P
i
,
n
N
{\displaystyle \displaystyle {A_{i}^{N}=\lim _{n\rightarrow +\infty }P_{i,n}^{N}}}
(*3-2)
c
a
r
d
Q
,
i
∈
C
0
(
P
o
l
y
t
o
p
e
s
i
(
R
N
)
,
F
)
{\displaystyle \displaystyle {{card}_{Q,i}\in {\mathcal {C}}^{0}{\Big (}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N}),F{\Big )}}}
c
a
r
d
Q
,
i
~
(
A
i
N
)
=
c
a
r
d
Q
,
i
~
(
lim
n
→
+
∞
P
i
,
n
N
)
=
lim
n
→
+
∞
c
a
r
d
Q
,
i
(
P
i
,
n
N
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,i}}}(A_{i}^{N})={\widetilde {{card}_{Q,i}}}(\lim _{n\rightarrow +\infty }P_{i,n}^{N})=\displaystyle {\lim _{n\rightarrow +\infty }{card}_{Q,i}(P_{i,n}^{N})}}}
et montrer que cette définition ne dépend pas de la suite
(
P
i
,
n
N
)
n
∈
N
{\displaystyle {(P_{i,n}^{N})}_{n\in \mathbb {N} }}
∀
n
∈
N
,
c
a
r
d
Q
,
i
(
P
i
,
n
N
)
=
∑
j
∈
N
i
c
j
,
i
(
P
i
,
n
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall n\in \mathbb {N} ,\,\,{card}_{Q,i}(P_{i,n}^{N})=\sum _{j\in \mathbb {N} _{i}}c_{j,i}(P_{i,n}^{N})\,\,{card}_{Q,1}^{j}([0,1[)}}
∃
!
c
a
r
d
Q
,
i
~
∈
C
0
(
P
V
i
(
R
N
)
,
F
)
{\displaystyle \displaystyle {\exists !{\widetilde {{card}_{Q,i}}}\in {\mathcal {C}}^{0}{\Big (}{PV}_{i}(\mathbb {R} ^{N}),F{\Big )}}}
telle que
∀
R
N
′
r
e
p
e
`
r
e
o
r
t
h
o
n
o
r
m
e
´
d
e
R
N
,
c
a
r
d
Q
,
R
N
′
|
P
V
i
(
R
N
)
=
c
a
r
d
Q
,
i
~
{\displaystyle \forall {{\cal {R}}_{N}}'\,\,rep{\grave {e}}re\,\,orthonorm{\acute {e}}\,\,de\,\,\mathbb {R} ^{N},\,\,{{card}_{Q,{{\cal {R}}_{N}}'}}_{{\Big |}{PV}_{i}(\mathbb {R} ^{N})}={\widetilde {{card}_{Q,i}}}}
et telle que
c
a
r
d
Q
,
i
~
|
P
o
l
y
t
o
p
e
s
i
(
R
N
)
=
c
a
r
d
Q
,
i
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,i}}}_{|{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}={card}_{Q,i}}}
[comme, on a (*1-2), (*2-2) et (*3-2)]
∀
i
∈
N
N
,
{\displaystyle \forall i\in \mathbb {N} _{N},}
∀
A
i
N
∈
P
V
i
(
R
N
)
,
{\displaystyle \displaystyle {\forall A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})},}
∃
(
P
i
,
n
N
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle \displaystyle {\exists {(P_{i,n}^{N})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}
A
i
N
=
lim
n
→
+
∞
P
i
,
n
N
,
{\displaystyle \displaystyle {A_{i}^{N}=\lim _{n\rightarrow +\infty }P_{i,n}^{N},}}
c
a
r
d
Q
,
i
~
(
A
i
N
)
=
c
a
r
d
Q
,
i
~
(
lim
n
→
+
∞
P
i
,
n
N
)
=
lim
n
→
+
∞
c
a
r
d
Q
,
i
(
P
i
,
n
N
)
=
lim
n
→
+
∞
∑
j
∈
N
i
c
j
,
i
(
P
i
,
n
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,i}}}(A_{i}^{N})={\widetilde {{card}_{Q,i}}}(\lim _{n\rightarrow +\infty }P_{i,n}^{N})=\lim _{n\rightarrow +\infty }{card}_{Q,i}(P_{i,n}^{N})=\lim _{n\rightarrow +\infty }\sum _{j\in \mathbb {N} _{i}}c_{j,i}(P_{i,n}^{N})\,\,{card}_{Q,1}^{j}([0,1[)}}
=
∑
j
∈
N
i
lim
n
→
+
∞
c
j
,
i
(
P
i
,
n
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
=
∑
j
∈
N
i
c
j
,
i
~
(
lim
n
→
+
∞
P
i
,
n
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
=
∑
j
∈
N
i
c
j
,
i
~
(
A
i
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
=
∑
j
∈
N
i
c
j
,
i
~
(
A
i
N
)
c
a
r
d
Q
,
1
~
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=\sum _{j\in \mathbb {N} _{i}}\lim _{n\rightarrow +\infty }c_{j,i}(P_{i,n}^{N})\,\,{card}_{Q,1}^{j}([0,1[)=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(\lim _{n\rightarrow +\infty }P_{i,n}^{N})\,\,{card}_{Q,1}^{j}([0,1[)=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(A_{i}^{N})\,\,{card}_{Q,1}^{j}([0,1[)=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(A_{i}^{N})\,\,{\widetilde {{card}_{Q,1}}}^{j}([0,1[)}}
∀
A
i
N
∈
P
V
i
(
R
N
)
,
c
a
r
d
Q
,
i
~
(
A
i
N
)
=
∑
j
∈
N
i
c
j
,
i
~
(
A
i
N
)
c
a
r
d
Q
,
1
~
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N}),\,\,{\widetilde {{card}_{Q,i}}}(A_{i}^{N})=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(A_{i}^{N})\,\,{\widetilde {{card}_{Q,1}}}^{j}([0,1[)}}
c
a
r
d
Q
,
i
~
:
P
V
i
(
R
N
)
⟶
F
:
A
i
N
↦
c
a
r
d
Q
,
i
~
(
A
i
N
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,i}}}\,\,:{PV}_{i}(\mathbb {R} ^{N})\,\,\longrightarrow \,\,F:\,\,A_{i}^{N}\,\,\mapsto \,\,{\widetilde {{card}_{Q,i}}}(A_{i}^{N})}}
c
a
r
d
Q
,
i
~
(
A
i
N
)
{\displaystyle {\widetilde {{card}_{Q,i}}}(A_{i}^{N})}
On peut aussi poser
c
a
r
d
Q
~
:
P
V
(
R
N
)
=
⨆
i
∈
N
N
P
V
i
(
R
N
)
⟶
F
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}:{PV}(\mathbb {R} ^{N})=\bigsqcup _{i\in \mathbb {N} _{N}}{PV}_{i}(\mathbb {R} ^{N})\longrightarrow F}}
telle que
c
a
r
d
Q
~
|
P
o
l
y
t
o
p
e
s
(
R
N
)
=
⨆
i
∈
N
N
P
o
l
y
t
o
p
e
s
i
(
R
N
)
=
c
a
r
d
Q
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}_{{\Big |}\displaystyle {{{\cal {P}}olytopes}(\mathbb {R} ^{N})=\bigsqcup _{i\in \mathbb {N} _{N}}{{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}={card_{Q}}}}
et telle que
∀
i
∈
N
N
,
c
a
r
d
Q
~
|
P
V
i
(
R
N
)
=
c
a
r
d
Q
,
i
~
{\displaystyle \displaystyle {\forall i\in \mathbb {N} _{N},\,\,{\widetilde {{card}_{Q}}}_{{\Big |}{PV}_{i}(\mathbb {R} ^{N})}={\widetilde {{card}_{Q,i}}}}}
[comme, on a (*1-2), (*2-2) et (*3-2)]
∀
i
∈
N
N
,
{\displaystyle \forall i\in \mathbb {N} _{N},}
∀
A
i
N
∈
P
V
i
(
R
N
)
,
{\displaystyle \displaystyle {\forall A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})},}
∃
(
P
i
,
n
N
)
n
∈
N
⊂
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle \displaystyle {\exists {(P_{i,n}^{N})}_{n\in \mathbb {N} }\subset {{\cal {P}}olytopes}_{i}(\mathbb {R} ^{N})}}
A
i
N
=
lim
n
→
+
∞
P
i
,
n
N
,
{\displaystyle \displaystyle {A_{i}^{N}=\lim _{n\rightarrow +\infty }P_{i,n}^{N},}}
c
a
r
d
Q
,
i
~
(
A
i
N
)
=
c
a
r
d
Q
,
i
~
(
lim
n
→
+
∞
P
i
,
n
N
)
=
lim
n
→
+
∞
c
a
r
d
Q
,
i
(
P
i
,
n
N
)
=
lim
n
→
+
∞
∑
j
∈
N
i
c
j
,
i
(
P
i
,
n
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q,i}}}(A_{i}^{N})={\widetilde {{card}_{Q,i}}}(\lim _{n\rightarrow +\infty }P_{i,n}^{N})=\lim _{n\rightarrow +\infty }{card}_{Q,i}(P_{i,n}^{N})=\lim _{n\rightarrow +\infty }\sum _{j\in \mathbb {N} _{i}}c_{j,i}(P_{i,n}^{N})\,\,{card}_{Q,1}^{j}([0,1[)}}
=
∑
j
∈
N
i
lim
n
→
+
∞
c
j
,
i
(
P
i
,
n
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
=
∑
j
∈
N
i
c
j
,
i
~
(
lim
n
→
+
∞
P
i
,
n
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
=
∑
j
∈
N
i
c
j
,
i
~
(
A
i
N
)
c
a
r
d
Q
,
1
j
(
[
0
,
1
[
)
=
∑
j
∈
N
i
c
j
,
i
~
(
A
i
N
)
c
a
r
d
Q
,
1
~
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=\sum _{j\in \mathbb {N} _{i}}\lim _{n\rightarrow +\infty }c_{j,i}(P_{i,n}^{N})\,\,{card}_{Q,1}^{j}([0,1[)=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(\lim _{n\rightarrow +\infty }P_{i,n}^{N})\,\,{card}_{Q,1}^{j}([0,1[)=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(A_{i}^{N})\,\,{card}_{Q,1}^{j}([0,1[)=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(A_{i}^{N})\,\,{\widetilde {{card}_{Q,1}}}^{j}([0,1[)}}
∀
A
i
N
∈
P
V
i
(
R
N
)
,
c
a
r
d
Q
,
i
~
(
A
i
N
)
=
∑
j
∈
N
i
c
j
,
i
~
(
A
i
N
)
c
a
r
d
Q
,
1
~
j
(
[
0
,
1
[
)
{\displaystyle \displaystyle {\forall A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N}),\,\,{\widetilde {{card}_{Q,i}}}(A_{i}^{N})=\sum _{j\in \mathbb {N} _{i}}{\widetilde {c_{j,i}}}(A_{i}^{N})\,\,{\widetilde {{card}_{Q,1}}}^{j}([0,1[)}}
Fin du théorème
Remarque :
Remarque :
F
{\displaystyle F}
N
⨆
+
∞
{\displaystyle \mathbb {N} \bigsqcup +\infty }
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle +\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
Remarque :
Le théorème précédent s'étend, très vraisemblablement, de manière analogue, aux parties compactes, convexes, (connexes) de
R
″
N
{\displaystyle {\mathbb {R} ''}^{N}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
C
1
{\displaystyle {\mathcal {C}}^{1}}
Début d’un théorème
Michel COSTE, dans ses PDF, a préféré dire que l'hypothèse de définition 3) avec les autres hypothèses de définition de la F-quantité impliquent que :
A
N
∈
P
V
N
(
R
N
)
{\displaystyle A_{N}\in {PV}_{N}(\mathbb {R} ^{N})}
C
1
{\displaystyle {\mathcal {C}}^{1}}
∀
n
∈
N
,
P
N
,
n
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle \forall n\in \mathbb {N} ,\,\,P_{N,n}\in {{\mathcal {P}}olytopes}_{N}(\mathbb {R} ^{N})}
lim
n
∈
N
,
n
→
+
∞
↑
P
N
,
n
=
A
N
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }\uparrow P_{N,n}=A_{N}}}
alors on a :
c
a
r
d
Q
~
(
A
N
)
=
c
a
r
d
Q
~
(
lim
n
∈
N
,
n
→
+
∞
P
N
,
n
)
=
lim
n
∈
N
,
n
→
+
∞
c
a
r
d
Q
(
P
N
,
n
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}(A_{N})={\widetilde {{card}_{Q}}}(\displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }P_{N,n}})=\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }{card}_{Q}(P_{N,n})}}
au lieu de dire qu'ils impliquent aussi, de manière plus faible, que :
A
N
∈
P
V
N
(
R
N
)
{\displaystyle A_{N}\in {PV}_{N}(\mathbb {R} ^{N})}
∀
n
∈
N
,
P
N
,
n
∈
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle \forall n\in \mathbb {N} ,\,\,P_{N,n}\in {{\mathcal {P}}olytopes}_{N}(\mathbb {R} ^{N})}
lim
n
∈
N
,
n
→
+
∞
↑
P
N
,
n
=
A
N
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }\uparrow P_{N,n}=A_{N}}}
alors on a :
c
a
r
d
Q
~
(
A
N
)
=
c
a
r
d
Q
~
(
lim
n
∈
N
,
n
→
+
∞
P
N
,
n
)
=
lim
n
∈
N
,
n
→
+
∞
c
a
r
d
Q
(
P
N
,
n
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}(A_{N})={\widetilde {{card}_{Q}}}(\displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }P_{N,n}})=\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }{card}_{Q}(P_{N,n})}}
Mais, de même, il aurait aussi préféré dire que cela implique que :
Si
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}
C
1
{\displaystyle {\mathcal {C}}^{1}}
∀
n
∈
N
,
P
i
,
n
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle \forall n\in \mathbb {N} ,\,\,P_{i,n}^{N}\in {{\mathcal {P}}olytopes}_{i}(\mathbb {R} ^{N})}
lim
n
∈
N
,
n
→
+
∞
↑
P
i
,
n
N
=
A
i
N
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }\uparrow P_{i,n}^{N}=A_{i}^{N}}}
alors on a :
c
a
r
d
Q
~
(
A
i
N
)
=
c
a
r
d
Q
~
(
lim
n
∈
N
,
n
→
+
∞
P
i
,
n
N
)
=
lim
n
∈
N
,
n
→
+
∞
c
a
r
d
Q
(
P
i
,
n
N
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}(A_{i}^{N})={\widetilde {{card}_{Q}}}(\displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }P_{i,n}^{N}})=\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }{card}_{Q}(P_{i,n}^{N})}}
au lieu de dire que cela implique aussi, de manière plus faible que :
Si
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
A
i
N
∈
P
V
i
(
R
N
)
{\displaystyle A_{i}^{N}\in {PV}_{i}(\mathbb {R} ^{N})}
∀
n
∈
N
,
P
i
,
n
N
∈
P
o
l
y
t
o
p
e
s
i
(
R
N
)
{\displaystyle \forall n\in \mathbb {N} ,\,\,P_{i,n}^{N}\in {{\mathcal {P}}olytopes}_{i}(\mathbb {R} ^{N})}
lim
n
∈
N
,
n
→
+
∞
↑
P
i
,
n
N
=
A
i
N
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }\uparrow P_{i,n}^{N}=A_{i}^{N}}}
alors on a :
c
a
r
d
Q
~
(
A
i
N
)
=
c
a
r
d
Q
~
(
lim
n
∈
N
,
n
→
+
∞
P
i
,
n
N
)
=
lim
n
∈
N
,
n
→
+
∞
c
a
r
d
Q
(
P
i
,
n
N
)
{\displaystyle \displaystyle {{\widetilde {{card}_{Q}}}(A_{i}^{N})={\widetilde {{card}_{Q}}}(\displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }P_{i,n}^{N}})=\lim _{n\in \mathbb {N} ,\,\,n\rightarrow +\infty }{card}_{Q}(P_{i,n}^{N})}}
Je tente de faire certaines généralisations.
Cela est, probablement, toujours, vrai,
si on remplace "
P
V
N
(
R
N
)
{\displaystyle {PV}_{N}(\mathbb {R} ^{N})}
par "
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
ou par "réunion finie de parties de
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
[et peut-être même, en supposant que
A
N
{\displaystyle A_{N}}
P
V
(
R
N
)
{\displaystyle {PV}(\mathbb {R} ^{N})}
∀
n
∈
N
,
P
N
,
n
{\displaystyle \forall n\in \mathbb {N} ,\,\,P_{N,n}}
P
o
l
y
t
o
p
e
s
N
(
R
N
)
{\displaystyle {{\mathcal {P}}olytopes}_{N}(\mathbb {R} ^{N})}
Si tel n'est pas le cas, il est facile de ramener le second cas au premier.
Fin du théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
∀
i
∈
N
N
∗
,
R
i
{\displaystyle \forall i\in \mathbb {N} _{N}^{*},\,\,{\cal {R}}_{i}}
R
i
{\displaystyle \mathbb {R} ^{i}}
O
i
(
0
)
j
∈
N
i
∗
{\displaystyle O_{i}{(0)}_{j\in \mathbb {N} _{i}^{*}}}
R
=
R
N
{\displaystyle {\cal {R}}={\cal {R}}_{N}}
On désigne par
∀
i
∈
N
N
∗
,
c
a
r
d
Q
,
i
=
c
a
r
d
Q
,
R
i
{\displaystyle \forall i\in \mathbb {N} _{N}^{*},\,\,{card}_{Q,i}={card}_{Q,{\cal {R}}_{i}}}
R
i
{\displaystyle {\cal {R}}_{i}}
c
a
r
d
Q
=
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q}={card}_{Q,{\cal {R}}}}
Remarque : La notion de F-quantité est une notion plus fine que celle de cardinal potentiel (ou de CANTOR) : Elle l'affine.
Mais, on ne sait pas, pour le moment, du moins concernant la partie connue et établie officiellement, aller au delà des parties d'une classe de parties bornées de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
Début d’un théorème
Soit
A
∈
P
(
R
)
{\displaystyle A\in {\cal {P}}(\mathbb {R} )}
Soient
f
:
A
⟶
R
{\displaystyle f:A\longrightarrow \mathbb {R} }
et
G
f
=
{
(
x
,
y
)
∈
A
×
R
|
y
=
f
(
x
)
}
{\displaystyle \displaystyle {G_{f}={\Big \{}(x,y)\in A\times \mathbb {R} {\Big |}y=f(x){\Big \}}}}
f
{\displaystyle f}
et
e
p
i
(
f
)
=
{
(
x
,
y
)
∈
A
×
R
|
y
≥
f
(
x
)
}
{\displaystyle {epi}(f)={\Big \{}(x,y)\in A\times \mathbb {R} {\Big |}y\geq f(x){\Big \}}}
f
{\displaystyle f}
f
(
A
)
{\displaystyle f(A)}
∀
a
∈
R
+
∗
,
c
a
r
d
Q
,
1
(
(
a
.
f
)
(
A
)
)
=
c
a
r
d
Q
,
1
(
f
(
A
)
)
{\displaystyle \forall a\in \mathbb {R} _{+}^{*},\,\,{card}_{Q,1}{\Big (}(a.f)(A){\Big )}={card}_{Q,1}{\Big (}f(A){\Big )}}
c
a
r
d
Q
,
1
(
(
0.
f
)
(
A
)
)
=
c
a
r
d
Q
,
1
(
{
0
}
)
=
1
≠
0
c
a
r
d
Q
,
1
(
f
(
A
)
)
=
0
{\displaystyle {card}_{Q,1}{\Big (}(0.f)(A){\Big )}={card}_{Q,1}(\{0\})=1\neq 0\,\,{card}_{Q,1}{\Big (}f(A){\Big )}=0}
c
a
r
d
Q
,
1
(
−
f
(
A
)
)
=
c
a
r
d
Q
,
1
(
f
(
A
)
)
{\displaystyle {card}_{Q,1}{\Big (}-f(A){\Big )}={card}_{Q,1}{\Big (}f(A){\Big )}}
f
,
g
:
A
⟶
R
{\displaystyle f,g\,\,:A\,\,\longrightarrow \mathbb {R} }
f
≤
g
⟹
e
p
i
(
f
)
⊃
e
p
i
(
g
)
⟹
c
a
r
d
Q
,
1
(
e
p
i
(
f
)
)
≥
c
a
r
d
Q
,
1
(
e
p
i
(
g
)
)
{\displaystyle f\leq g\Longrightarrow {epi}(f)\supset {epi}(g)\Longrightarrow {card}_{Q,1}{\Big (}{epi}(f){\Big )}\geq {card}_{Q,1}{\Big (}{epi}(g){\Big )}}
B
⊂
A
{\displaystyle B\subset A}
Comme
e
p
i
(
f
|
B
)
⊂
e
p
i
(
f
)
{\displaystyle epi(f_{|B})\subset {epi}(f)}
c
a
r
d
Q
,
1
(
e
p
i
(
f
|
B
)
)
≤
c
a
r
d
Q
,
1
(
e
p
i
(
f
)
)
{\displaystyle {card}_{Q,1}{\Big (}{epi}(f_{|B}){\Big )}\leq {card}_{Q,1}{\Big (}{epi}(f){\Big )}}
Fin du théorème
Début d’un théorème
Fin du théorème
Début d’un théorème
Soit
A
∈
P
(
R
)
{\displaystyle A\in {\cal {P}}(\mathbb {R} )}
∃
(
I
i
)
i
∈
Z
{\displaystyle \exists {(I_{i})}_{i\in \mathbb {Z} }}
A
{\displaystyle A}
∀
i
∈
Z
{\displaystyle \forall i\in \mathbb {Z} }
I
i
{\displaystyle I_{i}}
R
{\displaystyle \mathbb {R} }
R
{\displaystyle \mathbb {R} }
∅
{\displaystyle \emptyset }
Soit
f
∈
C
1
-
D
i
f
f
e
´
o
m
o
r
p
h
i
s
m
e
p
a
r
m
o
r
c
e
a
u
x
(
A
,
R
)
{\displaystyle f\in {{\mathcal {C}}^{1}}{\mbox{-}}{{\mathcal {D}}iff{\acute {e}}omorphisme\,\,par\,\,morceaux}(A,\mathbb {R} )}
c
a
r
d
Q
,
1
(
G
f
)
=
∑
i
∈
Z
c
a
r
d
Q
,
1
(
f
(
I
i
)
)
{\displaystyle \displaystyle {{card}_{Q,1}(G_{f})=\sum _{i\in \mathbb {Z} }{card}_{Q,1}{\Big (}f(I_{i}){\Big )}}}
Fin du théorème
R
n
{\displaystyle \mathbb {R} ^{n}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
R
n
{\displaystyle \mathbb {R} ^{n}}
n
=
2
{\displaystyle n=2}
[ modifier | modifier le wikicode ] Début d’un théorème
∀
n
∈
N
∗
,
∀
i
∈
N
n
,
c
a
r
d
Q
,
i
{\displaystyle \forall n\in \mathbb {N} ^{*},\,\,\forall i\in \mathbb {N} _{n},\,\,{card}_{Q,i}}
P
V
i
(
R
n
)
{\displaystyle {PV}_{i}(\mathbb {R} ^{n})}
∀
n
∈
N
∗
,
∀
i
∈
N
n
,
P
V
i
(
R
n
)
=
{
A
i
n
∈
P
V
(
R
n
)
|
d
i
m
(
A
i
n
)
=
i
}
⋃
{
∅
}
{\displaystyle \displaystyle {\forall n\in \mathbb {N} ^{*},\,\,\forall i\in \mathbb {N} _{n},\,\,{PV}_{i}(\mathbb {R} ^{n})=\{A_{i}^{n}\in {PV}(\mathbb {R} ^{n})\,\,|\,\,{dim}(A_{i}^{n})=i\}\bigcup \{\emptyset \}}}
donc :
c
a
r
d
Q
,
2
(
B
R
2
(
0
,
1
)
¯
)
{\displaystyle {card}_{Q,2}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(0,1)}}{\Big )}}
=
c
a
r
d
Q
,
2
(
⨆
x
∈
[
−
1
,
1
]
[
(
x
,
−
1
−
x
2
)
,
(
x
,
1
−
x
2
)
]
)
{\displaystyle \displaystyle {={card}_{Q,2}{\Bigg (}\bigsqcup _{x\in [-1,1]}{\bigg [}{\Big (}x,-{\sqrt {1-x^{2}}}{\Big )},{\Big (}x,{\sqrt {1-x^{2}}}{\Big )}{\bigg ]}{\Bigg )}}}
=
∫
[
−
1
,
1
]
c
a
r
d
Q
,
1
(
[
(
x
,
−
1
−
x
2
)
,
(
x
,
1
−
x
2
)
]
)
d
c
a
r
d
Q
,
1
(
x
)
{\displaystyle \displaystyle {=\int _{[-1,1]}{card}_{Q,1}{\Bigg (}{\bigg [}{\Big (}x,-{\sqrt {1-x^{2}}}{\Big )},{\Big (}x,{\sqrt {1-x^{2}}}{\Big )}{\bigg ]}{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)}}
=
∫
[
−
1
,
1
]
c
a
r
d
Q
,
1
(
[
−
1
−
x
2
,
1
−
x
2
]
)
d
c
a
r
d
Q
,
1
(
x
)
{\displaystyle \displaystyle {=\int _{[-1,1]}{card}_{Q,1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big ]}{\bigg )}\,\,d\,\,{card}_{Q,1}(x)}}
=
∫
]
−
1
,
1
[
c
a
r
d
Q
,
1
(
[
−
1
−
x
2
,
1
−
x
2
]
)
d
c
a
r
d
Q
,
1
(
x
)
+
2
c
a
r
d
Q
,
1
(
{
0
}
)
{\displaystyle \displaystyle {=\int _{]-1,1[}{card}_{Q,1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big ]}{\bigg )}\,\,d\,\,{card}_{Q,1}(x)+2\,\,{card}_{Q,1}(\{0\})}}
=
∫
]
−
1
,
1
[
(
c
a
r
d
Q
,
1
(
[
−
1
−
x
2
,
1
−
x
2
[
)
+
1
)
d
c
a
r
d
Q
,
1
(
x
)
+
2
{\displaystyle \displaystyle {=\int _{]-1,1[}{\Bigg (}{card}_{Q,1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big [}{\bigg )}+1{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)+2}}
=
∫
]
−
1
,
1
[
(
v
o
l
1
(
[
−
1
−
x
2
,
1
−
x
2
[
)
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
)
d
c
a
r
d
Q
,
1
(
x
)
+
2
{\displaystyle \displaystyle {=\int _{]-1,1[}{\Bigg (}{vol}^{1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big [}{\bigg )}\,\,{card}_{Q,1}([0,1[)+1{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)+2}}
=
∫
]
−
1
,
1
[
(
2
1
−
x
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
)
d
c
a
r
d
Q
,
1
(
x
)
+
2
{\displaystyle \displaystyle {=\int _{]-1,1[}{\Bigg (}2{\sqrt {1-x^{2}}}\,\,{card}_{Q,1}([0,1[)+1{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)+2}}
=
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
∫
]
−
1
,
1
[
d
c
a
r
d
Q
,
1
(
x
)
+
2
{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+\int _{]-1,1[}d\,\,{card}_{Q,1}(x)+2}}
=
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
c
a
r
d
Q
,
1
(
]
−
1
,
1
[
)
+
2
{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+{card}_{Q,1}(]-1,1[)+2}}
=
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
c
a
r
d
Q
,
1
(
[
−
1
,
1
[
)
−
1
+
2
{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+{card}_{Q,1}([-1,1[)-1+2}}
=
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
v
o
l
1
(
[
−
1
,
1
[
)
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+{vol}^{1}([-1,1[)\,\,{card}_{Q,1}([0,1[)+1}}
=
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+2\,\,{card}_{Q,1}([0,1[)+1}}
=
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
(
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
1
)
+
1
{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,{\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+1{\Big )}+1}}
c
a
r
d
Q
,
2
(
B
R
2
(
0
,
1
)
¯
)
=
π
c
a
r
d
Q
,
1
2
(
[
0
,
1
[
)
+
π
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
=
π
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
(
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
)
+
1
{\displaystyle \displaystyle {{card}_{Q,2}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(0,1)}}{\Big )}=\pi \,\,{card}_{Q,1}^{2}([0,1[)+\pi \,\,{card}_{Q,1}([0,1[)+1=\pi \,\,{card}_{Q,1}([0,1[)\,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}+1}}
donc
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
(
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
1
)
+
1
=
π
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
(
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
)
+
1
{\displaystyle \displaystyle {2\,\,{card}_{Q,1}([0,1[)\,\,{\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+1{\Big )}+1=\pi \,\,{card}_{Q,1}([0,1[)\,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}+1}}
c'est-à-dire
2
(
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
+
1
)
=
π
(
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
)
{\displaystyle \displaystyle {2\,\,{\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+1{\Big )}=\pi \,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}}}
c'est-à-dire
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
=
π
2
(
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
)
−
1
=
π
2
c
a
r
d
Q
,
1
(
[
0
,
1
]
)
−
1
{\displaystyle \displaystyle {\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)={\frac {\pi }{2}}\,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}-1={\frac {\pi }{2}}\,\,{card}_{Q,1}([0,1])-1}}
c'est-à-dire
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
=
(
∫
]
−
1
,
1
[
1
−
x
2
d
v
o
l
1
(
x
)
)
c
a
r
d
Q
,
1
(
[
0
,
1
]
)
−
1
{\displaystyle \displaystyle {\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)={\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{vol}^{1}(x){\Big )}\,\,{card}_{Q,1}([0,1])-1}}
c'est-à-dire
∫
]
−
1
,
1
[
1
−
x
2
d
c
a
r
d
Q
,
1
(
x
)
=
π
2
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
π
2
−
1
{\displaystyle \displaystyle {\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)={\frac {\pi }{2}}\,\,{card}_{Q,1}([0,1[)+{\frac {\pi }{2}}-1}}
Fin du théorème
Remarque :
]
−
1
,
1
[
∉
P
V
1
(
R
)
{\displaystyle ]-1,1[\not \in {PV}_{1}(\mathbb {R} )}
∀
n
∈
N
∗
,
∀
i
∈
N
n
,
{\displaystyle \forall n\in \mathbb {N} ^{*},\,\,\forall i\in \mathbb {N} _{n},\,\,}
c
a
r
d
Q
,
i
{\displaystyle {card}_{Q,i}}
P
V
i
(
R
n
)
{\displaystyle {PV}_{i}(\mathbb {R} ^{n})}
P
3
i
(
R
n
)
=
{
A
i
n
∈
P
(
R
n
)
|
A
i
n
b
o
r
n
e
´
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
e
t
d
i
m
(
A
i
n
)
=
i
}
{\displaystyle {P3}_{i}(\mathbb {R} ^{n})=\{A_{i}^{n}\in {\mathcal {P}}(\mathbb {R} ^{n})\,\,|\,\,A_{i}^{n}\,\,born{\acute {e}}e,\,\,convexe\,\,(connexe),\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,et\,\,{dim}(A_{i}^{n})=i\}}
(Calculs peut-être remis en cause car
c
a
r
d
Q
,
i
{\displaystyle {card}_{Q,i}}
P
V
i
(
R
n
)
{\displaystyle {PV}_{i}(\mathbb {R} ^{n})}
P
V
i
(
R
n
)
{\displaystyle {PV}_{i}(\mathbb {R} ^{n})}
Calcul de
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
{\displaystyle {card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}}
f
∈
C
1
-
d
i
f
f
e
´
o
m
o
r
p
h
i
s
m
e
(
A
¯
,
R
)
,
c
a
r
d
Q
,
1
-
m
e
s
u
r
a
b
l
e
{\displaystyle f\in {\mathcal {C}}^{1}{\mbox{-}}diff{\acute {e}}omorphisme({\overline {A}},\mathbb {R} ),\,\,{card}_{Q,1}{\mbox{-}}mesurable}
A
∈
P
3
(
R
)
{\displaystyle A\in {P3}(\mathbb {R} )}
[ modifier | modifier le wikicode ] Début d’un théorème
Remarque : Il y a peut-être des erreurs et des passages mal formulés voire faux.
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
Soit
i
∈
N
N
{\displaystyle i\in \mathbb {N} _{N}}
P
3
(
R
N
)
=
{
A
N
∈
P
(
R
N
)
|
A
N
c
o
n
v
e
x
e
(
s
i
m
p
l
e
m
e
n
t
c
o
n
n
e
x
e
)
,
b
o
r
n
e
´
e
d
e
R
N
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
}
{\displaystyle {P3}(\mathbb {R} ^{N})=\{A^{N}\in {\cal {P}}(\mathbb {R} ^{N})\,\,|\,\,A^{N}\,\,convexe\,\,(simplement\,\,connexe),\,\,born{\acute {e}}e\,\,de\,\,\mathbb {R} ^{N},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\}}
Soit
P
3
i
(
R
N
)
=
{
A
i
N
∈
P
(
R
N
)
|
A
i
N
c
o
n
v
e
x
e
(
s
i
m
p
l
e
m
e
n
t
c
o
n
n
e
x
e
)
,
b
o
r
n
e
´
e
d
e
R
N
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
e
t
d
i
m
(
A
i
N
)
=
i
}
{\displaystyle {P3}_{i}(\mathbb {R} ^{N})=\{A_{i}^{N}\in {\cal {P}}(\mathbb {R} ^{N})\,\,|\,\,A_{i}^{N}\,\,convexe\,\,(simplement\,\,connexe),\,\,born{\acute {e}}e\,\,de\,\,\mathbb {R} ^{N},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,et\,\,{dim}(A_{i}^{N})=i\}}
=
{
A
i
N
∈
P
3
(
R
N
)
|
d
i
m
(
A
i
N
)
=
i
}
{\displaystyle =\{A_{i}^{N}\in {P3}(\mathbb {R} ^{N})\,\,|\,\,{dim}(A_{i}^{N})=i\}}
A
N
=
A
N
N
∈
P
V
N
(
R
N
)
{\displaystyle A_{N}=A_{N}^{N}\in {PV}_{N}(\mathbb {R} ^{N})}
c
i
,
N
(
A
N
)
=
L
N
−
i
,
N
(
A
N
)
β
(
N
−
i
)
{\displaystyle \displaystyle {c_{i,N}(A_{N})={\frac {{\cal {L}}_{N-i,N}(A_{N})}{\beta (N-i)}}}}
où
β
(
i
)
=
v
o
l
i
(
B
R
i
(
O
i
,
1
)
¯
)
{\displaystyle \displaystyle {\beta (i)={vol}^{i}{\Big (}{\overline {B_{\mathbb {R} ^{i}}(O_{i},1)}}{\Big )}}}
O
i
{\displaystyle O_{i}}
R
i
{\displaystyle {\cal {R}}_{i}}
R
i
{\displaystyle \mathbb {R} ^{i}}
(
L
i
,
N
(
A
N
)
)
i
∈
N
N
{\displaystyle {{\Big (}{\mathcal {L}}_{i,N}(A_{N}){\Big )}}_{i\in \mathbb {N} _{N}}}
A
N
{\displaystyle A_{N}}
A
∈
P
3
i
(
R
N
)
{\displaystyle A\in {P3}_{i}(\mathbb {R} ^{N})}
A
¯
∈
P
V
i
(
R
N
)
{\displaystyle {\overline {A}}\in {PV}_{i}(\mathbb {R} ^{N})}
Soit
A
∈
P
3
(
R
N
)
{\displaystyle A\in {P3}(\mathbb {R} ^{N})}
A
¯
∈
P
V
(
R
N
)
{\displaystyle {\overline {A}}\in {PV}(\mathbb {R} ^{N})}
N
=
1
{\displaystyle N=1}
Soit
A
∈
P
3
1
(
R
)
=
P
3
(
R
)
{\displaystyle A\in {P3}_{1}(\mathbb {R} )={P3}(\mathbb {R} )}
A
¯
∈
P
V
1
(
R
)
=
P
V
(
R
)
{\displaystyle {\overline {A}}\in {PV}_{1}(\mathbb {R} )={PV}(\mathbb {R} )}
c
a
r
d
Q
,
1
(
A
¯
)
=
c
1
,
1
(
A
¯
)
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
c
0
,
1
(
A
¯
)
{\displaystyle \displaystyle {{card}_{Q,1}({\overline {A}})=c_{1,1}({\overline {A}})\,\,{card}_{Q,1}([0,1[)+c_{0,1}({\overline {A}})}}
f
:
R
⟶
R
,
c
a
r
d
Q
,
1
-
m
e
s
u
r
a
b
l
e
{\displaystyle f\,\,:\,\,\mathbb {R} \longrightarrow \mathbb {R} ,\,\,{card}_{Q,1}{\mbox{-}}mesurable}
∫
R
f
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
=
∫
R
f
(
x
)
d
(
c
1
,
1
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
c
0
,
1
)
(
x
)
=
∫
R
f
(
x
)
(
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
d
c
1
,
1
+
d
c
0
,
1
)
(
x
)
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
R
f
(
x
)
d
c
1
,
1
(
x
)
+
∫
R
f
(
x
)
d
c
0
,
1
(
x
)
{\displaystyle \displaystyle {\int _{\mathbb {R} }f(x)\,\,d\,\,{card}_{Q,1}(x)=\int _{\mathbb {R} }f(x)\,\,d\,\,{\Big (}c_{1,1}\,\,{card}_{Q,1}([0,1[)+c_{0,1}{\Big )}(x)=\int _{\mathbb {R} }f(x)\,\,{\Big (}{card}_{Q,1}([0,1[)\,\,d\,\,c_{1,1}+d\,\,c_{0,1}{\Big )}(x)={card}_{Q,1}([0,1[)\,\,\int _{\mathbb {R} }f(x)\,\,d\,\,c_{1,1}(x)+\int _{\mathbb {R} }f(x)\,\,d\,\,c_{0,1}(x)}}
B
∈
P
(
R
)
{\displaystyle B\in {\cal {P}}(\mathbb {R} )}
f
:
R
⟶
R
,
c
a
r
d
Q
,
1
-
m
e
s
u
r
a
b
l
e
{\displaystyle f\,\,:\,\,\mathbb {R} \longrightarrow \mathbb {R} ,\,\,{card}_{Q,1}{\mbox{-}}mesurable}
g
=
f
I
B
{\displaystyle g=f\,\,\mathbb {I} _{B}}
alors
∫
R
g
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
R
g
(
x
)
d
c
1
,
1
(
x
)
+
∫
R
g
(
x
)
d
c
0
,
1
(
x
)
{\displaystyle \displaystyle {\int _{\mathbb {R} }g(x)\,\,d\,\,{card}_{Q,1}(x)={card}_{Q,1}([0,1[)\,\,\int _{\mathbb {R} }g(x)\,\,d\,\,c_{1,1}(x)+\int _{\mathbb {R} }g(x)\,\,d\,\,c_{0,1}(x)}}
∫
R
(
f
I
B
)
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
R
(
f
I
B
)
(
x
)
d
c
1
,
1
(
x
)
+
∫
R
(
f
I
B
)
(
x
)
d
c
0
,
1
(
x
)
{\displaystyle \displaystyle {\int _{\mathbb {R} }(f\,\,\mathbb {I} _{B})(x)\,\,d\,\,{card}_{Q,1}(x)={card}_{Q,1}([0,1[)\,\,\int _{\mathbb {R} }(f\,\,\mathbb {I} _{B})(x)\,\,d\,\,c_{1,1}(x)+\int _{\mathbb {R} }(f\,\,\mathbb {I} _{B})(x)\,\,d\,\,c_{0,1}(x)}}
∫
B
f
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
B
f
(
x
)
d
c
1
,
1
(
x
)
+
∫
B
f
(
x
)
d
c
0
,
1
(
x
)
{\displaystyle \displaystyle {\int _{B}f(x)\,\,d\,\,{card}_{Q,1}(x)={card}_{Q,1}([0,1[)\,\,\int _{B}f(x)\,\,d\,\,c_{1,1}(x)+\int _{B}f(x)\,\,d\,\,c_{0,1}(x)}}
f
∈
C
1
-
d
i
f
f
e
´
o
m
o
r
p
h
i
s
m
e
(
A
¯
,
R
)
,
c
a
r
d
Q
,
1
-
m
e
s
u
r
a
b
l
e
{\displaystyle f\in {\mathcal {C}}^{1}{\mbox{-}}diff{\acute {e}}omorphisme({\overline {A}},\mathbb {R} ),\,\,{card}_{Q,1}{\mbox{-}}mesurable}
J
=
∫
A
¯
f
′
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
∫
A
¯
f
′
(
x
)
d
c
1
,
1
(
x
)
⏟
J
1
+
∫
A
¯
f
′
(
x
)
d
c
0
,
1
(
x
)
⏟
J
2
{\displaystyle \displaystyle {J=\int _{\overline {A}}f'(x)\,\,d\,\,{card}_{Q,1}(x)={card}_{Q,1}([0,1[)\,\,\underbrace {\int _{\overline {A}}f'(x)\,\,d\,\,c_{1,1}(x)} _{J_{1}}+\underbrace {\int _{\overline {A}}f'(x)\,\,d\,\,c_{0,1}(x)} _{J_{2}}}}
N
=
1
{\displaystyle N=1}
i
∈
N
N
=
N
1
{\displaystyle i\in \mathbb {N} _{N}=\mathbb {N} _{1}}
c
0
,
1
(
A
¯
)
=
L
1
,
1
(
A
¯
)
β
(
1
)
=
v
o
l
0
(
∂
A
¯
)
2
=
v
o
l
0
(
∂
A
)
2
{\displaystyle \displaystyle {c_{0,1}({\overline {A}})={\frac {{\cal {L}}_{1,1}({\overline {A}})}{\beta (1)}}={\frac {vol^{0}(\partial {\overline {A}})}{2}}={\frac {vol^{0}(\partial A)}{2}}}}
c
1
,
1
(
A
¯
)
=
L
0
,
1
(
A
¯
)
β
(
0
)
=
v
o
l
1
(
A
¯
)
1
=
v
o
l
1
(
A
¯
)
{\displaystyle \displaystyle {c_{1,1}({\overline {A}})={\frac {{\cal {L}}_{0,1}({\overline {A}})}{\beta (0)}}={\frac {{vol}^{1}({\overline {A}})}{1}}={vol}^{1}({\overline {A}})}}
J
1
=
∫
A
¯
f
′
(
x
)
d
c
1
,
1
(
x
)
=
∫
A
¯
f
′
(
x
)
d
v
o
l
1
(
x
)
=
∫
A
¯
d
v
o
l
1
(
f
(
x
)
)
=
∫
f
(
A
¯
)
d
v
o
l
1
(
x
)
=
v
o
l
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {J_{1}=\int _{\overline {A}}f'(x)\,\,d\,\,c_{1,1}(x)=\int _{\overline {A}}f'(x)\,\,d\,\,{vol}^{1}(x)=\int _{\overline {A}}d\,\,{vol}^{1}{\Big (}f(x){\Big )}=\int _{f({\overline {A}})}d\,\,{vol}^{1}(x)={vol}^{1}{\Big (}f({\overline {A}}){\Big )}}}
=
c
1
,
1
(
f
(
A
¯
)
)
{\displaystyle =c_{1,1}{\Big (}f({\overline {A}}){\Big )}}
J
2
=
∫
A
¯
f
′
(
x
)
d
c
0
,
1
(
x
)
=
∫
∂
A
f
′
(
x
)
d
v
o
l
0
(
x
)
2
=
1
2
∫
∂
A
f
′
(
x
)
d
v
o
l
0
(
x
)
{\displaystyle \displaystyle {J_{2}=\int _{\overline {A}}f'(x)\,\,d\,\,c_{0,1}(x)=\int _{\partial A}f'(x)\,\,d\,\,{\frac {vol^{0}(x)}{2}}={\frac {1}{2}}\,\,\int _{\partial A}f'(x)\,\,d\,\,vol^{0}(x)}}
A
¯
{\displaystyle {\overline {A}}}
R
{\displaystyle \mathbb {R} }
f
′
{\displaystyle f'}
A
¯
{\displaystyle {\overline {A}}}
f
′
|
A
¯
{\displaystyle {f'}_{|{\overline {A}}}}
∃
a
1
,
a
2
∈
A
¯
,
∂
A
=
{
a
1
,
a
2
}
{\displaystyle \exists a_{1},a_{2}\in {\overline {A}},\,\,\partial A=\{a_{1},a_{2}\}}
f
′
(
∂
A
)
=
{
f
′
(
a
1
)
,
f
′
(
a
2
)
}
{\displaystyle f'(\partial A)=\{f'(a_{1}),f'(a_{2})\}}
J
2
=
f
′
(
a
1
)
+
f
′
(
a
2
)
2
{\displaystyle \displaystyle {J_{2}={\frac {f'(a_{1})+f'(a_{2})}{2}}}}
c
0
,
1
(
f
(
A
¯
)
)
=
∫
f
(
A
¯
)
d
c
0
,
1
(
x
)
=
∫
A
¯
d
c
0
,
1
(
f
(
x
)
)
=
∫
∂
A
d
v
o
l
0
(
f
(
x
)
)
2
=
1
2
∫
∂
A
d
v
o
l
0
(
f
(
x
)
)
{\displaystyle \displaystyle {c_{0,1}{\Big (}f({\overline {A}}){\Big )}=\int _{f({\overline {A}})}\,\,d\,\,c_{0,1}(x)=\int _{\overline {A}}\,\,d\,\,c_{0,1}{\Big (}f(x){\Big )}=\int _{\partial A}d\,\,{\frac {vol^{0}{\Big (}f(x){\Big )}}{2}}={\frac {1}{2}}\,\,\int _{\partial A}d\,\,vol^{0}{\Big (}f(x){\Big )}}}
=
1
2
∫
f
(
∂
A
)
d
v
o
l
0
(
x
)
=
1
2
v
o
l
0
(
f
(
∂
A
)
)
=
1
2
×
2
=
1
{\displaystyle \displaystyle {={\frac {1}{2}}\,\,\int _{f(\partial A)}d\,\,vol^{0}(x)={\frac {1}{2}}\,\,vol^{0}{\Big (}f(\partial A){\Big )}={\frac {1}{2}}\times 2=1}}
A
¯
{\displaystyle {\overline {A}}}
R
{\displaystyle \mathbb {R} }
f
C
1
{\displaystyle f\,\,C^{1}}
A
¯
{\displaystyle {\overline {A}}}
A
¯
{\displaystyle {\overline {A}}}
f
|
A
¯
{\displaystyle f_{|{\overline {A}}}}
∂
A
=
{
a
1
,
a
2
}
{\displaystyle \partial A=\{a_{1},a_{2}\}}
f
(
∂
A
)
=
{
f
(
a
1
)
,
f
(
a
2
)
}
{\displaystyle f(\partial A)=\{f(a_{1}),f(a_{2})\}}
J
2
≠
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {J_{2}\neq c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
J
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
J
1
+
J
2
≠
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
c
1
,
1
(
f
(
A
¯
)
)
+
c
0
,
1
(
f
(
A
¯
)
)
=
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {J={card}_{Q,1}([0,1[)\,\,J_{1}+J_{2}\neq {card}_{Q,1}([0,1[)\,\,c_{1,1}{\Big (}f({\overline {A}}){\Big )}+c_{0,1}{\Big (}f({\overline {A}}){\Big )}={card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}}}
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
≠
∫
A
¯
f
′
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
{\displaystyle \displaystyle {{card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}\neq \int _{\overline {A}}f'(x)\,\,d\,\,{card}_{Q,1}(x)}}
J
2
=
(
∫
A
¯
f
′
(
x
)
d
c
0
,
1
(
x
)
)
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {J_{2}={\Big (}\int _{\overline {A}}f'(x)\,\,d\,\,c_{0,1}(x){\Big )}\,\,c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
∫
A
¯
f
′
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
{\displaystyle \displaystyle {\int _{\overline {A}}f'(x)\,\,d\,\,{card}_{Q,1}(x)}}
=
J
{\displaystyle =J}
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
J
1
+
J
2
{\displaystyle \displaystyle {={card}_{Q,1}([0,1[)\,\,J_{1}+J_{2}}}
=
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
c
1
,
1
(
f
(
A
¯
)
)
+
(
∫
A
¯
f
′
(
x
)
d
c
0
,
1
(
x
)
)
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {={card}_{Q,1}([0,1[)\,\,c_{1,1}{\Big (}f({\overline {A}}){\Big )}+{\Big (}\int _{\overline {A}}f'(x)\,\,d\,\,c_{0,1}(x){\Big )}\,\,c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
=
(
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
c
1
,
1
(
f
(
A
¯
)
)
+
c
0
,
1
(
f
(
A
¯
)
)
)
+
(
∫
A
¯
f
′
(
x
)
d
c
0
,
1
(
x
)
−
1
)
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {={\bigg (}{card}_{Q,1}([0,1[)\,\,c_{1,1}{\Big (}f({\overline {A}}){\Big )}+c_{0,1}{\Big (}f({\overline {A}}){\Big )}{\bigg )}+{\Big (}\int _{\overline {A}}f'(x)\,\,d\,\,c_{0,1}(x)-1{\Big )}\,\,c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
=
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
+
(
∫
A
¯
f
′
(
x
)
d
c
0
,
1
(
x
)
−
1
)
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {={card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}+{\Big (}\int _{\overline {A}}f'(x)\,\,d\,\,c_{0,1}(x)-1{\Big )}\,\,c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
=
∫
A
¯
f
′
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
−
(
∫
A
¯
f
′
(
x
)
d
c
0
,
1
(
x
)
−
1
)
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {{card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}=\int _{\overline {A}}f'(x)\,\,d\,\,{card}_{Q,1}(x)-{\Big (}\int _{\overline {A}}f'(x)\,\,d\,\,c_{0,1}(x)-1{\Big )}\,\,c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
=
∫
A
¯
f
′
(
x
)
d
c
a
r
d
Q
,
1
(
x
)
−
(
f
′
(
a
1
)
+
f
′
(
a
2
)
2
−
1
)
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {{card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}=\int _{\overline {A}}f'(x)\,\,d\,\,{card}_{Q,1}(x)-{\Big (}{\frac {f'(a_{1})+f'(a_{2})}{2}}-1{\Big )}\,\,c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
=
c
1
,
1
(
f
(
A
¯
)
)
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {{card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}=c_{1,1}{\Big (}f({\overline {A}}){\Big )}\,\,{card}_{Q,1}([0,1[)+c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
c
a
r
d
Q
(
f
(
A
¯
)
)
−
1
c
a
r
d
Q
,
1
(
[
0
,
1
]
)
−
1
=
v
o
l
1
(
f
(
A
¯
)
)
v
o
l
1
(
[
0
,
1
]
)
{\displaystyle \displaystyle {{\frac {{card}_{Q}{\Big (}f({\overline {A}}){\Big )}-1}{{card}_{Q,1}([0,1])-1}}={\frac {{vol}^{1}{\Big (}f({\overline {A}}){\Big )}}{{vol}^{1}([0,1])}}}}
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {{card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}}}
=
v
o
l
1
(
f
(
A
¯
)
)
v
o
l
1
(
[
0
,
1
]
)
c
a
r
d
Q
,
1
(
[
0
,
1
]
)
−
v
o
l
1
(
f
(
A
¯
)
)
v
o
l
1
(
[
0
,
1
]
)
+
1
{\displaystyle \displaystyle {={\frac {{vol}^{1}{\Big (}f({\overline {A}}){\Big )}}{{vol}^{1}([0,1])}}\,\,{card}_{Q,1}([0,1])-{\frac {{vol}^{1}{\Big (}f({\overline {A}}){\Big )}}{{vol}^{1}([0,1])}}+1}}
=
v
o
l
1
(
f
(
A
¯
)
)
c
a
r
d
Q
,
1
(
[
0
,
1
]
)
−
v
o
l
1
(
f
(
A
¯
)
)
+
1
{\displaystyle \displaystyle {={vol}^{1}{\Big (}f({\overline {A}}){\Big )}\,\,{card}_{Q,1}([0,1])-{vol}^{1}{\Big (}f({\overline {A}}){\Big )}+1}}
=
v
o
l
1
(
f
(
A
¯
)
)
(
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
)
−
v
o
l
1
(
f
(
A
¯
)
)
+
1
{\displaystyle \displaystyle {={vol}^{1}{\Big (}f({\overline {A}}){\Big )}\,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}-{vol}^{1}{\Big (}f({\overline {A}}){\Big )}+1}}
=
v
o
l
1
(
f
(
A
¯
)
)
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {={vol}^{1}{\Big (}f({\overline {A}}){\Big )}\,\,{card}_{Q,1}([0,1[)+1}}
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
=
v
o
l
1
(
f
(
A
¯
)
)
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {{card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}={vol}^{1}{\Big (}f({\overline {A}}){\Big )}\,\,{card}_{Q,1}([0,1[)+1}}
c
a
r
d
Q
,
1
(
f
(
A
¯
)
)
=
c
1
,
1
(
f
(
A
¯
)
)
c
a
r
d
Q
,
1
(
[
0
,
1
[
)
+
c
0
,
1
(
f
(
A
¯
)
)
{\displaystyle \displaystyle {{card}_{Q,1}{\Big (}f({\overline {A}}){\Big )}=c_{1,1}{\Big (}f({\overline {A}}){\Big )}\,\,{card}_{Q,1}([0,1[)+c_{0,1}{\Big (}f({\overline {A}}){\Big )}}}
Fin du théorème
R
N
{\displaystyle \mathbb {R} ^{N}}
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
N
∈
N
∗
{\displaystyle N\in \mathbb {N} ^{*}}
A
N
∈
P
(
R
N
)
{\displaystyle A_{N}\in {\cal {P}}(\mathbb {R} ^{N})}
R
N
{\displaystyle \mathbb {R} ^{N}}
N
{\displaystyle N}
Si
n
∈
N
N
{\displaystyle n\in \mathbb {N} _{N}}
″
∂
0
(
A
n
)
″
=
d
e
´
f
A
n
{\displaystyle {''\partial ^{0}(A_{n})''}{\underset {d{\acute {e}}f}{=}}A_{n}}
n
∈
N
N
∗
{\displaystyle n\in \mathbb {N} _{N}^{*}}
A
n
−
1
=
d
e
´
f
″
∂
1
(
A
n
)
″
{\displaystyle A_{n-1}{\underset {d{\acute {e}}f}{=}}{''\partial ^{1}(A_{n})''}}
A
n
{\displaystyle A_{n}}
A
n
−
1
{\displaystyle A_{n-1}}
R
n
{\displaystyle \mathbb {R} ^{n}}
n
−
1
{\displaystyle n-1}
(On pose
″
∂
1
(
A
N
)
″
=
d
e
´
f
A
N
∖
A
N
∘
{\displaystyle {''\partial ^{1}(A_{N})''}{\underset {d{\acute {e}}f}{=}}A_{N}\setminus {\stackrel {\circ }{A_{N}}}}
R
N
{\displaystyle \mathbb {R} ^{N}}
n
(
n
∈
N
N
−
1
)
{\displaystyle n\,\,(n\in \mathbb {N} _{N-1})}
et si
n
∈
N
N
∗
{\displaystyle n\in \mathbb {N} _{N}^{*}}
∀
i
∈
N
n
∗
{\displaystyle \forall i\in \mathbb {N} _{n}^{*}}
A
n
−
i
=
d
e
´
f
″
∂
i
(
A
n
)
″
=
d
e
´
f
″
∂
1
(
″
∂
i
−
1
(
A
n
)
″
)
″
{\displaystyle A_{n-i}{\underset {d{\acute {e}}f}{=}}{''\partial ^{i}(A_{n})''}{\underset {d{\acute {e}}f}{=}}{''\partial ^{1}{\Big (}{''\partial ^{i-1}(A_{n})''}{\Big )}''}}
A
n
−
i
{\displaystyle A_{n-i}}
R
n
{\displaystyle \mathbb {R} ^{n}}
n
−
i
{\displaystyle n-i}
A
0
{\displaystyle A_{0}}
A
N
=
[
⨆
i
∈
N
N
∗
(
A
N
−
(
i
−
1
)
∖
A
N
−
i
)
]
⨆
A
0
{\displaystyle \displaystyle {A_{N}=\left[\bigsqcup _{i\in \mathbb {N} _{N}^{*}}{\Big (}A_{N-(i-1)}\setminus A_{N-i}{\Big )}\right]\bigsqcup A_{0}}}
avec
∀
i
∈
N
N
∗
,
(
A
N
−
(
i
−
1
)
∖
A
N
−
i
)
réunion finie, disjointe, de sous-variétés, simplement connexes, ouvertes, de
R
N
, non vides, de dimension
N
−
(
i
−
1
)
{\displaystyle \forall i\in \mathbb {N} _{N}^{*},\,\,{\Big (}A_{N-(i-1)}\setminus A_{N-i}{\Big )}\,\,{\text{réunion finie, disjointe, de sous-variétés, simplement connexes, ouvertes, de}}\,\,\mathbb {R} ^{N}\,\,{\text{, non vides, de dimension}}\,\,N-(i-1)}
et
A
0
réunion finie, disjointe, de sous-variétés, simplement connexes, de
R
N
, non vides, de dimension
0
,
c'est-à-dire ensemble dénombrable
{\displaystyle A_{0}\,\,{\text{réunion finie, disjointe, de sous-variétés, simplement connexes, de}}\,\,\mathbb {R} ^{N}\,\,{\text{, non vides, de dimension}}\,\,0,\,\,{\text{c'est-à-dire ensemble dénombrable}}}
L'hébergeur de PDF gratuit utilisé ci-dessous (https://www.fichier-pdf.fr ) a été déclaré site fiable par FranceVerif, au moins, depuis le 11-10-2023.
https://www.fichier-pdf.fr/2014/06/16/decomposition-d-une-partie-bornee-de-r-2/
Fin du théorème
En fait, puisque la "F-quantité, relative au repère orthonormé
R
{\displaystyle {\mathcal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
{\displaystyle {\mathcal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
Q
F
,
R
(
A
)
{\displaystyle Q_{F,{\mathcal {R}}}(A)}
F
-
Q
R
(
A
)
{\displaystyle F{\text{-}}Q_{\mathcal {R}}(A)}
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q,{\mathcal {R}}}(A)}
F-quantité définie sur
P
V
(
R
n
)
⨆
P
V
2
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle {PV}({\mathbb {R} }^{n})\bigsqcup {PV2}({\mathbb {R} }^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
I
{\displaystyle I}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ]
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
[
A
,
(
A
i
)
i
∈
I
]
∈
P
(
R
n
)
×
F
(
I
,
P
(
R
n
)
)
{\displaystyle [A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}(\mathbb {R} ^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
I
{\displaystyle I}
(Si de plus,
I
{\displaystyle I}
sup
(
I
)
=
n
o
t
+
∞
I
{\displaystyle \sup(I){\underset {not}{=}}+\infty _{I}}
Soit
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
Soit
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
A
i
=
A
{\displaystyle \displaystyle {{\underset {i\in I,\,\,i\rightarrow \sup(I)}{{\text{lim}}_{classique}}}A_{i}=A}}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
[
A
,
(
A
i
)
i
∈
I
]
∈
P
(
R
n
)
×
F
(
I
,
P
(
R
n
)
)
{\displaystyle {\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\in {\mathcal {P}}(\mathbb {R} ^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}}}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
A
i
=
A
⇔
d
e
´
f
(
⋃
i
∈
I
A
i
=
A
s
i
(
A
i
)
i
∈
I
↗
p
o
u
r
⊂
,
o
u
,
⋂
i
∈
I
A
i
=
A
s
i
(
A
i
)
i
∈
I
↘
p
o
u
r
⊂
)
{\displaystyle \displaystyle {{\underset {i\in I,\,\,i\rightarrow \sup(I)}{{\text{lim}}_{classique}}}A_{i}=A\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,{\Big (}\bigcup _{i\in I}A_{i}=A\,\,si\,\,{(A_{i})}_{i\in I}\,\,\nearrow \,\,pour\,\,\subset ,\,\,ou,\,\,\bigcap _{i\in I}A_{i}=A\,\,si\,\,{(A_{i})}_{i\in I}\,\,\searrow \,\,pour\,\,\subset {\Big )}}}
alors que la définition et la notation non classiques de limite d'une famille de parties
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
n
{\displaystyle \mathbb {R} ^{n}}
[
A
,
(
A
i
)
i
∈
I
]
∈
P
(
R
n
)
×
F
(
I
,
P
(
R
n
)
)
{\displaystyle {\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\in {\mathcal {P}}(\mathbb {R} ^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
⇔
d
e
´
f
(
⋃
i
∈
I
A
i
=
A
s
i
(
A
i
)
i
∈
I
↗
p
o
u
r
⊂
,
o
u
,
⋂
i
∈
I
A
i
=
A
s
i
(
A
i
)
i
∈
I
↘
p
o
u
r
⊂
)
{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,{\Big (}\bigcup _{i\in I}A_{i}=A\,\,si\,\,{(A_{i})}_{i\in I}\,\,\nearrow \,\,pour\,\,\subset ,\,\,ou,\,\,\bigcap _{i\in I}A_{i}=A\,\,si\,\,{(A_{i})}_{i\in I}\,\,\searrow \,\,pour\,\,\subset {\Big )}}}
[
A
,
(
A
i
)
i
∈
I
]
∈
P
(
R
n
)
×
F
(
I
,
P
(
R
n
)
)
{\displaystyle [A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}({\mathbb {R} }^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}({\mathbb {R} }^{n}){\Big )}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
A
i
=
A
⇔
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle \displaystyle {{\underset {i\in I,\,\,i\rightarrow \sup(I)}{{\text{lim}}_{classique}}}A_{i}=A\,\,\Leftrightarrow \,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}}}
NB : Ce changement de notion et de notation n'est pas sans conséquences.
Fin du théorème
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
A
,
B
)
{\displaystyle {{\mathcal {P}}lafonnements}(I,{\mathcal {A}},{\mathcal {B}})}
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
A
)
{\displaystyle {{\mathcal {P}}lafonnements}(I,{\mathcal {A}})}
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
A
,
B
∈
P
2
(
R
n
)
{\displaystyle {\mathcal {A}},{\mathcal {B}}\in {\mathcal {P}}^{2}(\mathbb {R} ^{n})}
I
{\displaystyle I}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
Soit
I
{\displaystyle I}
Soient
A
,
B
∈
P
2
(
R
n
)
{\displaystyle {\mathcal {A}},{\mathcal {B}}\in {\mathcal {P}}^{2}(\mathbb {R} ^{n})}
On pose :
P
b
o
r
n
e
´
e
s
(
R
n
)
=
d
e
´
f
{
A
∈
P
(
R
n
)
|
A
b
o
r
n
e
´
e
d
a
n
s
R
n
}
{\displaystyle {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n}){\underset {d{\acute {e}}f}{=}}\{A\in {\mathcal {P}}(\mathbb {R} ^{n})\,\,|\,\,A\,\,born{\acute {e}}e\,\,dans\,\,\mathbb {R} ^{n}\}}
On pose :
P
n
o
n
b
o
r
n
e
´
e
s
(
R
n
)
=
d
e
´
f
{
A
∈
P
(
R
n
)
|
A
n
o
n
b
o
r
n
e
´
e
d
a
n
s
R
n
}
{\displaystyle {\mathcal {P}}_{non\,\,born{\acute {e}}es}(\mathbb {R} ^{n}){\underset {d{\acute {e}}f}{=}}\{A\in {\mathcal {P}}(\mathbb {R} ^{n})\,\,|\,\,A\,\,non\,\,born{\acute {e}}e\,\,dans\,\,\mathbb {R} ^{n}\}}
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
A
,
B
)
=
d
e
´
f
{
[
A
,
(
A
i
)
i
∈
I
]
∈
A
×
F
(
I
,
B
)
|
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
}
{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}(I,{\mathcal {A}},{\mathcal {B}}){\underset {d{\acute {e}}f}{=}}{\Big \{}[A,{(A_{i})}_{i\in I}]\in {\mathcal {A}}\times {\mathcal {F}}(I,{\mathcal {B}})\,\,{\Big |}\,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}=[A,{(A_{i})}_{i\in I}]{\Big \}}}}
On pose :
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
A
)
=
d
e
´
f
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
A
,
A
)
=
d
e
´
f
{
[
A
,
(
A
i
)
i
∈
I
]
∈
A
×
F
(
I
,
A
)
|
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
}
{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}(I,{\mathcal {A}}){\underset {d{\acute {e}}f}{=}}{{\mathcal {P}}lafonnements}(I,{\mathcal {A}},{\mathcal {A}}){\underset {d{\acute {e}}f}{=}}{\Big \{}[A,{(A_{i})}_{i\in I}]\in {\mathcal {A}}\times {\mathcal {F}}(I,{\mathcal {A}})\,\,{\Big |}\,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}=[A,{(A_{i})}_{i\in I}]{\Big \}}}}
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
=
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
,
P
(
R
n
)
)
=
{
[
A
,
(
A
i
)
i
∈
I
]
∈
P
(
R
n
)
×
F
(
I
,
P
(
R
n
)
)
|
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
}
{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}={{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}),{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}={\Big \{}[A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}(\mathbb {R} ^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}\,\,{\Big |}\,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}=[A,{(A_{i})}_{i\in I}]{\Big \}}}}
Fin du théorème
Motivation :
"Hypothèse de définition ou Conjecture, concernant la F-quantité, impliquant un plafonnement "
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
A
∈
P
V
2
(
R
n
)
{\displaystyle A\in {PV2}(\mathbb {R} ^{n})}
(
A
i
)
i
∈
I
⊂
P
V
(
R
n
)
{\displaystyle {(A_{i})}_{i\in I}\subset {PV}(\mathbb {R} ^{n})}
A
∈
P
4
(
R
n
)
{\displaystyle A\in {P4}(\mathbb {R} ^{n})}
(
A
i
)
i
∈
I
⊂
P
3
(
R
n
)
{\displaystyle {(A_{i})}_{i\in I}\subset {P3}(\mathbb {R} ^{n})}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
et qui servira
dans "Propositions concernant certains intervalles
I
{\displaystyle I}
R
{\displaystyle \mathbb {R} }
P
V
2
(
R
)
{\displaystyle {PV2}(\mathbb {R} )}
dans "Exemples illustratifs de calculs, avec la F-quantité/Exemples 2" ,
dans "Exemples illustratifs de calculs, avec la F-quantité/2 calculs de la F-quantité de
R
2
{\displaystyle \mathbb {R} ^{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
O
2
(
0
,
0
)
{\displaystyle O_{2}(0,0)}
R
2
{\displaystyle {\cal {R}}_{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
dans "Exemples illustratifs de calculs, avec la F-quantité/Plafonnement sphérique, {associé à|de}
R
n
{\displaystyle \mathbb {R} ^{n}}
O
n
{\displaystyle O_{n}}
R
n
{\displaystyle {\cal {R_{n}}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
et dans "Autres tentatives de généralisation de la F-quantité sur
R
n
{\displaystyle \mathbb {R} ^{n}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
.
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
P
4
(
R
n
)
{\displaystyle {P4}(\mathbb {R} ^{n})}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
P
V
2
(
R
n
)
{\displaystyle {PV2}(\mathbb {R} ^{n})}
=
d
e
´
f
{
A
∈
P
(
R
n
)
|
A
s
o
u
s
-
v
a
r
i
e
´
t
e
´
f
e
r
m
e
´
e
,
n
o
n
b
o
r
n
e
´
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
d
e
R
n
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
,
o
u
s
a
n
s
b
o
r
d
}
{\displaystyle {\underset {d{\acute {e}}f}{=}}{\Big \{}{A}\in {\cal {P}}(\mathbb {R} ^{n})\,\,{\Big |}\,\,{A}\,\,sous{\text{-}}vari{\acute {e}}t{\acute {e}}\,\,ferm{\acute {e}}e,\,\,non\,\,born{\acute {e}}e,\,\,convexe\,\,(connexe)\,\,de\,\,\mathbb {R} ^{n},\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux),\,\,ou\,\,sans\,\,bord{\Big \}}}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
=
d
e
´
f
{
A
∈
P
(
R
n
)
|
A
b
o
r
n
e
´
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
o
u
s
a
n
s
b
o
r
d
}
{\displaystyle {\underset {d{\acute {e}}f}{=}}{\Big \{}A\in {\mathcal {P}}(\mathbb {R} ^{n})\,\,|\,\,A\,\,born{\acute {e}}e,\,\,convexe\,\,(connexe),\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,ou\,\,sans\,\,bord{\Big \}}}
et
P
4
(
R
n
)
{\displaystyle {P4}(\mathbb {R} ^{n})}
=
d
e
´
f
{
A
∈
P
(
R
n
)
|
A
n
o
n
b
o
r
n
e
´
e
,
c
o
n
v
e
x
e
(
c
o
n
n
e
x
e
)
,
d
e
c
l
a
s
s
e
(
C
0
)
e
t
(
C
1
p
a
r
m
o
r
c
e
a
u
x
)
o
u
s
a
n
s
b
o
r
d
}
{\displaystyle {\underset {d{\acute {e}}f}{=}}{\Big \{}A\in {\mathcal {P}}(\mathbb {R} ^{n})\,\,|\,\,A\,\,non\,\,born{\acute {e}}e,\,\,convexe\,\,(connexe),\,\,de\,\,classe\,\,({\mathcal {C}}^{0})\,\,et\,\,({\mathcal {C}}^{1}\,\,par\,\,morceaux)\,\,ou\,\,sans\,\,bord{\Big \}}}
Fin du théorème
P
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
I
{\displaystyle I}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
Soit
I
{\displaystyle I}
Soit
R
{\displaystyle {\mathcal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
c
a
r
d
Q
,
R
:
P
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
→
N
⨆
+
∞
{\displaystyle {card}_{Q,{\mathcal {R}}}\,\,:\,\,{\mathcal {P}}(\mathbb {R} ^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}\,\,\rightarrow \,\,\mathbb {N} \bigsqcup +\infty }
P
(
R
n
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})}
P
b
o
r
n
e
´
e
s
(
R
n
)
{\displaystyle {\mathcal {P}}_{born{\acute {e}}es}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
de manière non classique et naïve , on considère
+
∞
{\displaystyle +\infty }
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle \{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
Fin du théorème
P
V
(
R
n
)
⨆
P
V
2
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle {PV}(\mathbb {R} ^{n})\bigsqcup {PV2}(\mathbb {R} ^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
I
{\displaystyle I}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
Soit
I
{\displaystyle I}
Soit
R
{\displaystyle {\mathcal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
c
a
r
d
Q
,
R
|
P
V
(
R
n
)
⨆
P
V
2
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
:
P
V
(
R
n
)
⨆
P
V
2
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
→
N
⨆
+
∞
:
A
↦
c
a
r
d
Q
,
R
(
A
)
=
?
{\displaystyle \displaystyle {{{card}_{Q,{\mathcal {R}}}}_{\displaystyle {{\Big |}{PV}(\mathbb {R} ^{n})\bigsqcup {PV2}(\mathbb {R} ^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}}\,\,:\,\,{PV}(\mathbb {R} ^{n})\bigsqcup {PV2}(\mathbb {R} ^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}\,\,\rightarrow \,\,\mathbb {N} \bigsqcup +\infty \,\,:\,\,A\,\,\mapsto \,\,{card}_{Q,{\mathcal {R}}}(A)=?}}
c
a
r
d
Q
,
R
|
P
V
(
R
n
)
=
c
a
r
d
Q
~
{\displaystyle \displaystyle {{{card}_{Q,{\mathcal {R}}}}_{\displaystyle {{\Big |}{PV}(\mathbb {R} ^{n})}}={\widetilde {{card}_{Q}}}}}
P
(
R
n
)
{\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
de manière non classique , on considère : "
+
∞
{\displaystyle +\infty }
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle +\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
Fin du théorème
Remarque :
P
V
(
R
n
)
⨆
P
V
2
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle {PV}(\mathbb {R} ^{n})\bigsqcup {PV2}(\mathbb {R} ^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
P
3
(
R
n
)
⨆
P
4
(
R
n
)
⨆
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
4
(
R
n
)
,
P
3
(
R
n
)
)
{\displaystyle \displaystyle {{P3}(\mathbb {R} ^{n})\bigsqcup {P4}(\mathbb {R} ^{n})\bigsqcup {{\mathcal {P}}lafonnements}{\Big (}I,{P4}(\mathbb {R} ^{n}),{P3}(\mathbb {R} ^{n}){\Big )}}}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
A
∈
P
V
2
(
R
n
)
{\displaystyle A\in {PV2}(\mathbb {R} ^{n})}
(
A
i
)
i
∈
I
⊂
P
V
(
R
n
)
{\displaystyle {(A_{i})}_{i\in I}\subset {PV}(\mathbb {R} ^{n})}
A
∈
P
4
(
R
n
)
{\displaystyle A\in {P4}(\mathbb {R} ^{n})}
(
A
i
)
i
∈
I
⊂
P
3
(
R
n
)
{\displaystyle {(A_{i})}_{i\in I}\subset {P3}(\mathbb {R} ^{n})}
I
{\displaystyle I}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
I
{\displaystyle I}
et si
A
∈
P
V
2
(
R
n
)
{\displaystyle A\in {PV2}(\mathbb {R} ^{n})}
A
{\displaystyle A}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
[ou peut-être même en supposant seulement que :
A
∈
P
4
(
R
n
)
{\displaystyle A\in {P4}(\mathbb {R} ^{n})}
A
{\displaystyle A}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
[ou peut-être même en supposant seulement que les parties de cette famille sont des parties de
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle \displaystyle {\lim _{i\in I,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
)
=
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
{\displaystyle card_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}=card_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}card_{Q,{\cal {R}}}(A_{i})}
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle {\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
Fin du théorème
Remarque :
Il se peut que si le résultat précédent est une conjecture, que, pour le démontrer, il faille admettre, comme hypothèses de définition, des cas particuliers de cette conjecture, impliquant des familles de parties
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
R
{\displaystyle \mathbb {R} }
R
{\displaystyle \mathbb {R} }
Remarque :
Questions :
Pour toute partie de
P
4
(
R
n
)
{\displaystyle {P4}(\mathbb {R} ^{n})}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
P
4
(
R
n
)
{\displaystyle {P4}(\mathbb {R} ^{n})}
Pour toute réunion infinie dénombrable disjointe de parties de
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
Pour toute partie de
P
4
(
R
n
)
{\displaystyle {P4}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
4
(
R
n
)
{\displaystyle {P4}(\mathbb {R} ^{n})}
Pour toute réunion infinie dénombrable disjointe de parties de
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
3
(
R
n
)
{\displaystyle {P3}(\mathbb {R} ^{n})}
Motivation principale de l'Hypothèse de définition ou de la Conjecture :
Soit
A
∈
P
V
2
(
R
n
)
{\displaystyle A\in {PV2}({\mathbb {R} }^{n})}
Soit
(
A
i
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}}
P
V
(
R
n
)
{\displaystyle {PV}({\mathbb {R} }^{n})}
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
=
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle \displaystyle {\lim _{i\in I,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}}}
Soit
(
B
i
)
i
∈
I
{\displaystyle {(B_{i})}_{i\in I}}
P
V
(
R
n
)
{\displaystyle {PV}({\mathbb {R} }^{n})}
telle que
(
B
i
)
i
∈
I
≠
(
A
i
)
i
∈
I
{\displaystyle {(B_{i})}_{i\in I}\neq {(A_{i})}_{i\in I}}
et telle que
lim
i
∈
I
,
i
→
sup
(
I
)
B
i
=
[
A
,
(
B
i
)
i
∈
I
]
{\displaystyle \displaystyle {\lim _{i\in I,i\rightarrow \sup(I)}B_{i}={\Big [}A,{(B_{i})}_{i\in I}{\Big ]}}}
c'est-à-dire telle que :
[
A
,
(
A
i
)
i
∈
I
]
≠
[
A
,
(
B
i
)
i
∈
I
]
{\displaystyle {\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\neq {\Big [}A,{(B_{i})}_{i\in I}{\Big ]}}
Si l'on suppose, de plus, que :
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
≠
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
B
i
)
{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})\neq \lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
)
=
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
≠
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
B
i
)
=
c
a
r
d
Q
,
R
(
lim
i
∈
I
,
i
→
sup
(
I
)
B
i
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
B
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})\neq \lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}B_{i})={card}_{Q,{\cal {R}}}{\Big (}[A,(B_{i})_{i\in I}]{\Big )}}
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
=
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
B
i
)
{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
)
=
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
=
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
B
i
)
=
c
a
r
d
Q
,
R
(
lim
i
∈
I
,
i
→
sup
(
I
)
B
i
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
B
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}B_{i})={card}_{Q,{\cal {R}}}{\Big (}[A,(B_{i})_{i\in I}]{\Big )}}
On aurait :
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
A
i
=
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
B
i
=
A
{\displaystyle \displaystyle {{\underset {i\in I,\,\,i\to \sup(I)}{{\text{lim}}_{classique}}}A_{i}={\underset {i\in I,\,\,i\to \sup(I)}{{\text{lim}}_{classique}}}B_{i}=A}}
et en supposant, de plus, que :
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
≠
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
B
i
)
{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})\neq \lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})}}
on aurait :
c
a
r
d
Q
,
R
(
A
)
=
c
a
r
d
Q
,
R
(
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
A
i
)
=
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
≠
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
B
i
)
=
c
a
r
d
Q
,
R
(
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
B
i
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q,{\cal {R}}}(A)={card}_{Q,{\cal {R}}}({\underset {i\in I,\,\,i\to \sup(I)}{{\text{lim}}_{classique}}}A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})\neq \lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})={card}_{Q,{\cal {R}}}({\underset {i\in I,\,\,i\to \sup(I)}{{\text{lim}}_{classique}}}B_{i})={card}_{Q,{\cal {R}}}(A)}
c'est-à-dire une contradiction.
@Attention : Sans précisions supplémentaires, l'expression "
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})}}
c
a
r
d
Q
,
R
(
lim
c
l
a
s
s
i
q
u
e
i
∈
I
,
i
→
sup
(
I
)
A
i
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({\underset {i\in I,\,\,i\to \sup(I)}{{\text{lim}}_{classique}}}A_{i})}}
c
a
r
d
Q
,
R
(
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
Il faut donc peut-être distinguer ces 2 usages, en distinguant les 2 notions de limite.
Il y a peut-être incompatibilité entre ces 2 notions de limite.
On peut, toutefois, dans un premier temps, définir, pour toutes les parties de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
Le problème est que l'on définit la F-quantité, relative à un repère orthonormé, de chacun des plafonnements de chaque partie de
R
n
{\displaystyle \mathbb {R} ^{n}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
lim
i
∈
I
,
i
→
sup
(
I
)
A
i
)
{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})}
lim
i
∈
I
,
i
→
sup
(
I
)
c
a
r
d
Q
,
R
(
A
i
)
{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})}}
R
n
{\displaystyle \mathbb {R} ^{n}}
A
∈
P
V
2
(
R
n
)
{\displaystyle A\in {PV2}({\mathbb {R} }^{n})}
(
A
i
)
i
∈
I
⊂
P
V
(
R
n
)
{\displaystyle {(A_{i})}_{i\in I}\subset {PV}({\mathbb {R} }^{n})}
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle [A,(A_{i})_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}({\mathbb {R} }^{n}){\Big )}}
avec
P
V
2
(
R
n
)
,
P
V
(
R
n
)
∈
P
2
(
R
n
)
{\displaystyle {PV2}({\mathbb {R} }^{n}),{PV}({\mathbb {R} }^{n})\in {\mathcal {P}}^{2}({\mathbb {R} }^{n})}
P
V
2
(
R
n
)
⋂
P
V
(
R
n
)
=
∅
{\displaystyle \displaystyle {{PV2}({\mathbb {R} }^{n})\bigcap {PV}({\mathbb {R} }^{n})=\emptyset }}
P
V
(
R
n
)
¯
P
V
2
(
R
n
)
=
P
V
2
(
R
n
)
{\displaystyle {\overline {{PV}({\mathbb {R} }^{n})}}^{{PV2}({\mathbb {R} }^{n})}={PV2}({\mathbb {R} }^{n})}
Plus généralement, on peut choisir
A
∈
A
{\displaystyle A\in {\mathcal {A}}}
(
A
i
)
i
∈
I
⊂
B
{\displaystyle {(A_{i})}_{i\in I}\subset {\mathcal {B}}}
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle [A,(A_{i})_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}({\mathbb {R} }^{n}){\Big )}}
avec
A
,
B
∈
P
2
(
R
n
)
{\displaystyle {\mathcal {A}},{\mathcal {B}}\in {\mathcal {P}}^{2}({\mathbb {R} }^{n})}
A
⋂
B
=
∅
{\displaystyle \displaystyle {{\mathcal {A}}\bigcap {\mathcal {B}}=\emptyset }}
B
¯
A
=
A
{\displaystyle {\overline {\mathcal {B}}}^{\mathcal {A}}={\mathcal {A}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
et étant donné un plafonnement normal non trivial de la partie
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,(A_{i})_{i\in I}]}
alors on peut définir la F-quantité de la partie
A
{\displaystyle A}
R
n
{\displaystyle \mathbb {R} ^{n}}
c
a
r
d
Q
,
R
(
A
)
=
d
e
´
f
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\cal {R}}}(A){\underset {d{\acute {e}}f}{=}}{card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}}
Conjecture qui servira :
dans "Propositions concernant certains intervalles
I
{\displaystyle I}
R
{\displaystyle \mathbb {R} }
P
V
2
(
R
)
{\displaystyle {PV2}(\mathbb {R} )}
dans "Exemples illustratifs de calculs, avec la F-quantité/Exemples 2" ,
dans "Exemples illustratifs de calculs, avec la F-quantité/2 calculs de la F-quantité de
R
2
{\displaystyle \mathbb {R} ^{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
O
2
(
0
,
0
)
{\displaystyle O_{2}(0,0)}
R
2
{\displaystyle {\cal {R}}_{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
dans "Exemples illustratifs de calculs, avec la F-quantité/Plafonnement sphérique, {associé à|de}
R
n
{\displaystyle \mathbb {R} ^{n}}
O
n
{\displaystyle O_{n}}
R
n
{\displaystyle {\cal {R_{n}}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
et dans "Autres tentatives de généralisation de la F-quantité sur
R
n
{\displaystyle \mathbb {R} ^{n}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
.
Remarque :
Je ne sais pas si j'ai justifié, suffisamment, convenablement et proprement, ces nouvelles notations, mais l'idée est là.
Au lieu de vouloir, toujours, exiger et demander, des conditions trop fortes concernant la notion dont il est question, peut-être faut-il, parfois, les affaiblir et accepter et se contenter de ces dernières, dans leurs versions affaiblies.
De toute façon, ce qu'on perd n'est rien en comparaison de ce qu'on gagne par ailleurs.
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
I
{\displaystyle I}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
Soit
I
{\displaystyle I}
Soit
R
{\displaystyle {\cal {R}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
O
{\displaystyle O}
[a)
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
≥
0
{\displaystyle card_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}])\geq 0}
b)
∀
[
∅
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
{
∅
}
,
P
(
R
n
)
)
{\displaystyle \forall [\emptyset ,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,\{\emptyset \},{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
c
a
r
d
Q
,
R
(
[
∅
,
(
A
i
)
i
∈
I
]
)
=
0
{\displaystyle {card}_{Q,{\cal {R}}}([\emptyset ,{(A_{i})}_{i\in I}])=0}
c)
∀
x
∈
R
n
,
∀
[
{
x
}
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
{
{
x
}
}
,
P
(
R
n
)
)
{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,\forall [\{x\},{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\Big \{}\{x\}{\Big \}},{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
c
a
r
d
Q
,
R
(
[
{
x
}
,
(
A
i
)
i
∈
I
]
)
=
1
{\displaystyle {card}_{Q,{\cal {R}}}([\{x\},{(A_{i})}_{i\in I}])=1}
∀
[
A
,
(
A
i
)
i
∈
I
]
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
[
A
⋃
B
,
(
A
i
⋃
B
i
)
i
∈
I
]
,
[
A
⋂
B
,
(
A
i
⋂
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \displaystyle {{\Big [}A\bigcup B,{{\Big (}A_{i}\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]},{\Big [}A\bigcap B,{{\Big (}A_{i}\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}}
et
∀
[
A
,
(
A
i
)
i
∈
I
]
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
c
a
r
d
Q
,
R
(
[
A
⋃
B
,
(
A
i
⋃
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
+
c
a
r
d
Q
,
R
(
[
B
,
(
B
i
)
i
∈
I
]
)
−
c
a
r
d
Q
,
R
(
[
A
⋂
B
,
(
A
i
⋂
B
i
)
i
∈
I
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A\bigcup B,{{\Big (}A_{i}\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}={card}_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}])+{card}_{Q,{\cal {R}}}([B,{(B_{i})}_{i\in I}])-{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A\bigcap B,{{\Big (}A_{i}\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}}}
et on pose :
∀
[
A
,
(
A
i
)
i
∈
I
]
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
[
A
,
(
A
i
)
i
∈
I
]
⋃
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
⋃
B
,
(
A
i
⋃
B
i
)
i
∈
I
]
{\displaystyle \displaystyle {{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcup B,{{\Big (}A_{i}\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]}}}
et
[
A
,
(
A
i
)
i
∈
I
]
⋂
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
⋂
B
,
(
A
i
⋂
B
i
)
i
∈
I
]
{\displaystyle \displaystyle {{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcap B,{{\Big (}A_{i}\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}}}
et donc on a :
∀
[
A
,
(
A
i
)
i
∈
I
]
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
⋃
[
B
,
(
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
+
c
a
r
d
Q
,
R
(
[
B
,
(
B
i
)
i
∈
I
]
)
−
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
⋂
[
B
,
(
B
i
)
i
∈
I
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\Big )}={card}_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}])+{card}_{Q,{\cal {R}}}([B,{(B_{i})}_{i\in I}])-{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\Big )}}}
et on pose :
∀
[
A
,
(
A
)
i
∈
I
]
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A)}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
A
⋃
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
,
(
A
)
i
∈
I
]
⋃
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
⋃
B
,
(
A
⋃
B
i
)
i
∈
I
]
{\displaystyle A\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A,{(A)}_{i\in I}{\Big ]}\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcup B,{{\Big (}A\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]}}
et
A
⋂
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
,
(
A
)
i
∈
I
]
⋂
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
⋂
B
,
(
A
⋂
B
i
)
i
∈
I
]
{\displaystyle A\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A,{(A)}_{i\in I}{\Big ]}\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcap B,{{\Big (}A\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}}
∀
(
x
m
)
m
∈
N
⊂
R
,
c
o
n
v
e
r
g
e
n
t
e
d
a
n
s
R
,
lim
m
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
x
m
]
)
=
c
a
r
d
Q
,
R
(
lim
m
→
sup
(
N
)
[
0
,
x
m
]
)
{\displaystyle \displaystyle {\forall {(x_{m})}_{m\in \mathbb {N} }\subset \mathbb {R} ,\,\,convergente\,\,dans\,\,\mathbb {R} ,\,\,\lim _{m\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,x_{m}])={card}_{Q,{\cal {R}}}(\lim _{m\rightarrow \sup(\mathbb {N} )}[0,x_{m}])}}
Soient
∀
i
∈
N
n
∗
,
R
i
{\displaystyle \forall i\in \mathbb {N} _{n}^{*},\,\,{\cal {R}}_{i}}
R
i
{\displaystyle {\mathbb {R} }^{i}}
O
i
(
0
)
j
∈
N
i
∗
{\displaystyle O_{i}{(0)}_{j\in \mathbb {N} _{i}^{*}}}
∀
k
∈
N
n
−
1
,
{\displaystyle \forall k\in \mathbb {N} _{n-1},}
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
−
k
)
)
{\displaystyle \displaystyle {\forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n-k}){\Big )}}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
k
)
)
{\displaystyle \displaystyle {\forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{k}){\Big )}}}
[
A
×
B
,
(
A
i
×
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \displaystyle {[A\times B,{(A_{i}\times B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}}
et
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
−
k
)
,
P
V
(
R
n
−
k
)
)
{\displaystyle \displaystyle {\forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n-k}),{PV}(\mathbb {R} ^{n-k}){\Big )}}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
k
)
,
P
V
(
R
k
)
)
{\displaystyle \displaystyle {\forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{k}),{PV}(\mathbb {R} ^{k}){\Big )}}}
c
a
r
d
Q
,
R
(
[
A
×
B
,
(
A
i
×
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
n
(
[
A
×
B
,
(
A
i
×
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
n
−
k
(
[
A
,
(
A
i
)
i
∈
I
]
)
c
a
r
d
Q
,
R
k
(
[
B
,
(
B
i
)
i
∈
I
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([A\times B,{(A_{i}\times B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n}}([A\times B,{(A_{i}\times B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n-k}}([A,{(A_{i})}_{i\in I}])\,\,{card}_{Q,{\cal {R}}_{k}}([B,{(B_{i})}_{i\in I}])}}
et on pose :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
−
k
)
)
{\displaystyle \displaystyle {\forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n-k}){\Big )}}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
k
)
)
{\displaystyle \displaystyle {\forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{k}){\Big )}}}
[
A
,
(
A
i
)
i
∈
I
]
×
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
×
B
,
(
A
i
×
B
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]\times [B,{(B_{i})}_{i\in I}]{\underset {d{\acute {e}}f}{=}}[A\times B,{(A_{i}\times B_{i})}_{i\in I}]}
et donc on a :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
−
k
)
,
P
V
(
R
n
−
k
)
)
{\displaystyle \displaystyle {\forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n-k}),{PV}(\mathbb {R} ^{n-k}){\Big )}}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
k
)
,
P
V
(
R
k
)
)
{\displaystyle \displaystyle {\forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{k}),{PV}(\mathbb {R} ^{k}){\Big )}}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
×
[
B
,
(
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
n
(
[
A
,
(
A
i
)
i
∈
I
]
×
[
B
,
(
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
n
−
k
(
[
A
,
(
A
i
)
i
∈
I
]
)
c
a
r
d
Q
,
R
k
(
[
B
,
(
B
i
)
i
∈
I
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}]\times [B,{(B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n}}([A,{(A_{i})}_{i\in I}]\times [B,{(B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n-k}}([A,{(A_{i})}_{i\in I}])\,\,{card}_{Q,{\cal {R}}_{k}}([B,{(B_{i})}_{i\in I}])}}
∀
[
A
,
(
A
i
)
i
∈
I
]
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
(
[
A
,
(
A
i
)
i
∈
I
]
⊂
[
B
,
(
B
i
)
i
∈
I
]
)
⇔
d
e
´
f
(
A
⊂
B
e
t
∀
i
∈
I
,
A
i
⊂
B
i
)
{\displaystyle \displaystyle {{\Big (}[A,{(A_{i})}_{i\in I}]\subset [B,{(B_{i})}_{i\in I}]{\Big )}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,(A\subset B\,\,et\,\,\forall i\in I,\,\,A_{i}\subset B_{i})}}
∀
[
A
,
(
A
i
)
i
∈
I
]
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
(
[
A
,
(
A
i
)
i
∈
I
]
∈
P
(
[
B
,
(
B
i
)
i
∈
I
]
)
)
⇔
d
e
´
f
(
A
∈
P
(
B
)
e
t
(
A
i
)
i
∈
I
∈
∏
i
∈
I
P
(
B
i
)
)
⇔
d
e
´
f
(
A
∈
P
(
B
)
e
t
∀
i
∈
I
,
A
i
∈
P
(
B
i
)
)
{\displaystyle \displaystyle {{\bigg (}[A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[B,{(B_{i})}_{i\in I}]{\Big )}{\bigg )}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,{\bigg (}A\in {\mathcal {P}}(B)\,\,et\,\,{(A_{i})}_{i\in I}\in \prod _{i\in I}{\mathcal {P}}(B_{i}){\bigg )}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,{\Big (}A\in {\mathcal {P}}(B)\,\,et\,\,\forall i\in I,\,\,A_{i}\in {\mathcal {P}}(B_{i}){\Big )}}}
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
[
A
∖
B
,
(
A
i
∖
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle [A\setminus B,{(A_{i}\setminus B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
Alors :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
c
a
r
d
Q
,
R
(
[
A
∖
B
,
(
A
i
∖
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
−
c
a
r
d
Q
,
R
(
[
B
,
(
B
i
)
i
∈
I
]
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A\setminus B,{(A_{i}\setminus B_{i})}_{i\in I}])={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B_{i})}_{i\in I}])}}
On pose :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
[
A
,
(
A
i
)
i
∈
I
]
∖
[
B
,
(
B
i
)
i
∈
I
]
=
d
e
´
f
[
A
∖
B
,
(
A
i
∖
B
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]\setminus [B,{(B_{i})}_{i\in I}]{\underset {d{\acute {e}}f}{=}}[A\setminus B,{(A_{i}\setminus B_{i})}_{i\in I}]}
On a donc :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
,
[
B
,
(
B
i
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
∖
[
B
,
(
B
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
−
c
a
r
d
Q
,
R
(
[
B
,
(
B
i
)
i
∈
I
]
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]\setminus [B,{(B_{i})}_{i\in I}])={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B_{i})}_{i\in I}])}}
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
,
[
B
,
(
B
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B)}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
[
A
∖
B
,
(
A
i
∖
B
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle [A\setminus B,{(A_{i}\setminus B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
Alors :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
(
R
n
)
,
P
V
(
R
n
)
)
,
[
B
,
(
B
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B)}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
c
a
r
d
Q
,
R
(
[
A
∖
B
,
(
A
i
∖
B
)
]
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
−
c
a
r
d
Q
,
R
(
[
B
,
(
B
)
i
∈
I
]
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A\setminus B,{(A_{i}\setminus B)]}_{i\in I}])={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B)}_{i\in I}])}}
On pose :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
,
[
B
,
(
B
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B)}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
[
A
,
(
A
i
)
i
∈
I
]
∖
B
=
d
e
´
f
[
A
,
(
A
i
)
i
∈
I
]
∖
[
B
,
(
B
)
i
∈
I
]
=
d
e
´
f
[
A
∖
B
,
(
A
i
∖
B
)
]
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]\setminus B{\underset {d{\acute {e}}f}{=}}[A,{(A_{i})}_{i\in I}]\setminus [B,{(B)}_{i\in I}]{\underset {d{\acute {e}}f}{=}}[A\setminus B,{(A_{i}\setminus B)]}_{i\in I}]}
On a donc :
∀
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
2
(
R
n
)
,
P
V
(
R
n
)
)
{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}
∀
[
B
,
(
B
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
V
(
R
n
)
,
P
V
(
R
n
)
)
,
[
B
,
(
B
)
i
∈
I
]
∈
P
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle \forall [B,{(B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B)}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
∖
B
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]\setminus B)}}
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
∖
[
B
,
(
B
)
i
∈
I
]
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]\setminus [B,{(B)}_{i\in I}])}}
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
−
c
a
r
d
Q
,
R
(
[
B
,
(
B
)
i
∈
I
]
)
{\displaystyle ={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B)}_{i\in I}])}
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
−
c
a
r
d
Q
,
R
(
B
)
{\displaystyle ={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}(B)}
Fin du théorème
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
I
{\displaystyle I}
Soit
A
∈
P
(
R
n
)
{\displaystyle A\in {\mathcal {P}}(\mathbb {R} ^{n})}
∃
(
A
i
)
i
∈
I
∈
{
(
A
i
)
i
∈
I
∈
F
(
I
,
P
(
R
n
)
)
|
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
}
{\displaystyle \exists {(A_{i})}_{i\in I}\in {\Big \{}{(A_{i})}_{i\in I}\in {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}\,\,{\Big |}\,\,[A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}{\Big \}}}
c
a
r
d
Q
,
R
(
A
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\mathcal {R}}}(A)={card}_{Q,{\mathcal {R}}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
A
{\displaystyle A}
(
A
i
)
i
∈
I
=
(
A
)
i
∈
I
{\displaystyle {(A_{i})}_{i\in I}={(A)}_{i\in I}}
[
A
,
(
A
i
)
i
∈
I
]
=
[
A
,
(
A
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]=[A,{(A)}_{i\in I}]}
A
{\displaystyle A}
De même, si
I
{\displaystyle I}
A
sup
(
I
)
=
A
max
(
I
)
=
A
{\displaystyle A_{\sup(I)}=A_{\max(I)}=A}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
A
{\displaystyle A}
On pose
P
l
a
f
o
n
n
e
m
e
n
t
s
n
o
r
m
a
u
x
(
I
,
P
(
R
n
)
)
=
d
e
´
f
{
[
A
,
(
A
i
)
i
∈
I
]
∈
P
l
a
f
o
n
n
e
m
e
n
t
s
(
I
,
P
(
R
n
)
)
|
c
a
r
d
Q
,
R
(
A
)
=
c
a
r
d
Q
,
R
(
[
A
,
(
A
i
)
i
∈
I
]
)
}
{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}_{normaux}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}{\underset {d{\acute {e}}f}{=}}{\Big \{}[A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}\,\,{\Big |}\,\,{card}_{Q,{\mathcal {R}}}(A)={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]){\Big \}}}}
Fin du théorème
Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
Soit
R
=
R
n
{\displaystyle {\mathcal {R}}={\mathcal {R}}_{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
O
n
{\displaystyle O_{n}}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
P
(
R
n
)
{\displaystyle {\cal {P}}({\mathbb {R} ^{n}})}
σ
{\displaystyle \sigma }
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
σ
{\displaystyle \sigma }
P
(
R
n
)
{\displaystyle {\cal {P}}({\mathbb {R} }^{n})}
n
=
1
{\displaystyle n=1}
R
+
=
⨆
i
∈
N
∗
[
i
−
1
,
i
[
et
R
+
=
⨆
i
∈
N
∗
[
2
i
−
2
,
2
i
[
{\displaystyle \displaystyle {{\mathbb {R} }_{+}=\bigsqcup _{i\in {\mathbb {N} }^{*}}[i-1,i[\,\,{\mbox{et}}\,\,{\mathbb {R} }_{+}=\bigsqcup _{i\in {\mathbb {N} }^{*}}[2i-2,2i[}}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
σ
{\displaystyle \sigma }
c
a
r
d
Q
,
R
(
R
+
)
=
c
a
r
d
Q
,
R
(
⨆
i
∈
N
∗
[
i
−
1
,
i
[
)
=
∑
i
∈
N
∗
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
=
∑
i
∈
N
∗
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({\mathbb {R} }_{+})={card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in {\mathbb {N} }^{*}}[i-1,i[{\Big )}=\sum _{i\in {\mathbb {N} }^{*}}{card}_{Q,{\cal {R}}}([i-1,i[)=\sum _{i\in {\mathbb {N} }^{*}}{card}_{Q,{\cal {R}}}([0,1[)}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
∑
i
∈
N
∗
1
=
c
a
r
d
Q
,
R
(
N
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {\mathbb {N} }^{*}}1={card}_{Q,{\cal {R}}}({\mathbb {N} }^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
c
a
r
d
Q
,
R
(
R
+
)
=
c
a
r
d
Q
,
R
(
⨆
i
∈
N
∗
[
2
i
−
2
,
2
i
[
)
=
∑
i
∈
N
∗
c
a
r
d
Q
,
R
(
[
2
i
−
2
,
2
i
[
)
=
∑
i
∈
N
∗
c
a
r
d
Q
,
R
(
[
0
,
2
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({\mathbb {R} }_{+})={card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in {\mathbb {N} }^{*}}[2i-2,2i[{\Big )}=\sum _{i\in {\mathbb {N} }^{*}}{card}_{Q,{\cal {R}}}([2i-2,2i[)=\sum _{i\in {\mathbb {N} }^{*}}{card}_{Q,{\cal {R}}}([0,2[)}}
=
∑
i
∈
N
∗
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
∑
i
∈
N
∗
1
=
2
c
a
r
d
Q
,
R
(
N
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=\sum _{i\in {\mathbb {N} }^{*}}2\,\,{card}_{Q,{\cal {R}}}([0,1[)=2\,\,{card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {\mathbb {N} }^{*}}1=2\,\,{card}_{Q,{\cal {R}}}({\mathbb {N} }^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
c
a
r
d
Q
,
R
(
N
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
≠
2
c
a
r
d
Q
,
R
(
N
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle {card}_{Q,{\cal {R}}}({\mathbb {N} }^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)\neq 2\,\,{card}_{Q,{\cal {R}}}({\mathbb {N} }^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}
c
a
r
d
Q
,
R
(
R
+
)
≠
c
a
r
d
Q
,
R
(
R
+
)
{\displaystyle {card}_{Q,{\cal {R}}}({\mathbb {R} }_{+})\neq {card}_{Q,{\cal {R}}}({\mathbb {R} }_{+})}
c
a
r
d
Q
,
R
{\displaystyle {card}_{Q,{\cal {R}}}}
σ
{\displaystyle \sigma }
donc ce n'est pas une mesure au sens usuel.
Les résultats seront différents suivant le choix des plafonnements de
R
{\displaystyle \mathbb {R} }
O
{\displaystyle O}
R
{\displaystyle {\cal {R}}}
R
{\displaystyle \mathbb {R} }
Réinterprétons les calculs ci-dessus, avec de nouvelles notions et notations :
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle +\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
sup
(
N
)
=
sup
(
R
)
=
+
∞
N
=
+
∞
R
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \sup(\mathbb {N} )=\sup(\mathbb {R} )=+\infty _{\mathbb {N} }=+\infty _{\mathbb {R} }=+\infty _{classique}}
R
1
,
+
=
[
R
+
,
(
[
0
,
r
]
)
r
∈
N
]
{\displaystyle R_{1,+}={\Big [}\mathbb {R} _{+},{([0,r])}_{r\in \mathbb {N} }{\Big ]}}
R
2
,
+
=
[
R
+
,
(
[
0
,
r
[
)
r
∈
N
]
{\displaystyle R_{2,+}={\Big [}\mathbb {R} _{+},{([0,r[)}_{r\in \mathbb {N} }{\Big ]}}
R
1
=
[
R
+
,
(
[
−
r
,
r
]
)
r
∈
N
]
{\displaystyle R_{1}={\Big [}\mathbb {R} _{+},{([-r,r])}_{r\in \mathbb {N} }{\Big ]}}
R
2
=
[
R
+
,
(
]
−
r
,
r
[
)
r
∈
N
]
{\displaystyle R_{2}={\Big [}\mathbb {R} _{+},{(]-r,r[)}_{r\in \mathbb {N} }{\Big ]}}
N
1
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
{\displaystyle \displaystyle {N_{1}={\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}}
N
1
∗
=
N
1
∖
{
0
}
{\displaystyle \displaystyle {N_{1}^{*}=N_{1}\setminus \{0\}}}
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
=
d
e
´
f
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
n
o
r
m
a
l
,
R
(
N
)
=
d
e
´
f
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
1
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
{\displaystyle \displaystyle {\sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} ){\underset {d{\acute {e}}f}{=}}\sup _{non\,\,classique,\,\,normal,\,\,{\mathcal {R}}}(\mathbb {N} ){\underset {d{\acute {e}}f}{=}}\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1})={card}_{Q,{\mathcal {R}}}(N_{1}^{*})}}
=
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
R
)
=
d
e
´
f
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
n
o
r
m
a
l
,
R
(
R
)
=
d
e
´
f
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
R
2
)
=
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
R
2
,
+
)
∈
+
∞
{\displaystyle \displaystyle {=\sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {R} ){\underset {d{\acute {e}}f}{=}}\sup _{non\,\,classique,\,\,normal,\,\,{\mathcal {R}}}(\mathbb {R} ){\underset {d{\acute {e}}f}{=}}\sup _{non\,\,classique,\,\,{\mathcal {R}}}(R_{2})=\sup _{non\,\,classique,\,\,{\mathcal {R}}}(R_{2,+})\in +\infty }}
c
a
r
d
Q
,
R
(
R
2
,
+
)
=
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
[
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
p
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{2,+})={card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,p[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p[)}}
•(1)
=
lim
p
∈
N
,
p
→
sup
(
N
)
p
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
lim
p
∈
N
,
p
→
sup
(
N
)
p
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p\,\,{card}_{Q,{\cal {R}}}([0,1[)={card}_{Q,{\cal {R}}}([0,1[)\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p}}
•(2)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
i
−
1
,
i
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[i-1,i[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[)=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([i-1,i[)}}
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[)={card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1}}
•[point où se rejoignent (1) et (2)]
=
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
2
p
[
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
2
p
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
2
p
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,2p[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,2p[)}}
•(1)
=
lim
p
∈
N
,
p
→
sup
(
N
)
2
p
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
lim
p
∈
N
,
p
→
sup
(
N
)
p
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}2p\,\,{card}_{Q,{\cal {R}}}([0,1[)=2\,\,{card}_{Q,{\cal {R}}}([0,1[)\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p}}
•(2)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
2
i
−
2
,
2
i
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
2
i
−
2
,
2
i
[
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[2i-2,2i[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([2i-2,2i[)}}
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
2
i
−
2
,
2
i
[
)
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
0
,
2
[
)
{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([2i-2,2i[)=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,2[)}}
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}2\,\,{card}_{Q,{\cal {R}}}([0,1[)=2\,\,{card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1}}
•[point où se rejoignent (1) et (2)]
=
2
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
=
2
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=2\,\,{card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
Or
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
≠
2
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle {card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)\neq 2\,\,{card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}
et donc
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
[
)
p
∈
N
]
)
≠
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
2
p
[
)
p
∈
N
]
)
{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}\neq {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
2
p
[
)
p
∈
N
]
)
=
2
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
[
)
p
∈
N
]
)
{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}
On a bien
(
[
0
,
p
[
)
p
∈
N
≠
(
[
0
,
2
p
[
)
p
∈
N
{\displaystyle {([0,p[)}_{p\in \mathbb {N} }\neq {([0,2p[)}_{p\in \mathbb {N} }}
Et on a :
c
a
r
d
Q
,
R
(
R
1
,
+
)
=
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
]
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
p
]
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
p
]
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
p
[
⨆
{
p
}
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{1,+})={card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,p]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p])=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p[\bigsqcup \{p\})}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
+
c
a
r
d
Q
,
R
(
{
p
}
)
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}{card}_{Q,{\cal {R}}}([0,p[)+{card}_{Q,{\cal {R}}}(\{p\}){\Big )}}}
•(1)
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
+
1
)
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
)
+
1
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\,\,{\Big (}{card}_{Q,{\cal {R}}}([0,p[)+1{\Big )}={\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\,\,{card}_{Q,{\cal {R}}}([0,p[){\Big )}+1}}
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
p
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
)
+
1
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
lim
p
∈
N
,
p
→
sup
(
N
)
p
)
+
1
{\displaystyle \displaystyle {={\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p\,\,{card}_{Q,{\cal {R}}}([0,1[){\Big )}+1={card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p{\Big )}+1}}
•(2)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
i
−
1
,
i
[
)
⨆
{
p
}
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
)
+
c
a
r
d
Q
,
R
(
{
p
}
)
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}\left(\left(\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[i-1,i[\right)\bigsqcup \{p\}\right)=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+{card}_{Q,{\cal {R}}}(\{p\}){\bigg )}}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
)
+
1
)
=
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
)
+
1
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1{\bigg )}={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1}}
=
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
)
+
1
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
)
+
1
{\displaystyle \displaystyle {={\bigg (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[){\bigg )}+1={card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\Big )}+1}}
•[point où se rejoignent (1) et (2)]
=
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
2
p
]
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
2
p
]
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
2
p
]
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
2
p
[
⨆
{
2
p
}
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,2p]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,2p])=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,2p[\bigsqcup \{2p\})}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
c
a
r
d
Q
,
R
(
[
0
,
2
p
[
)
+
c
a
r
d
Q
,
R
(
{
2
p
}
)
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}{card}_{Q,{\cal {R}}}([0,2p[)+{card}_{Q,{\cal {R}}}(\{2p\}){\Big )}}}
•(1)
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
c
a
r
d
Q
,
R
(
[
0
,
2
p
[
)
+
1
)
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
2
p
[
)
)
+
1
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\,\,{\Big (}{card}_{Q,{\cal {R}}}([0,2p[)+1{\Big )}={\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\,\,{card}_{Q,{\cal {R}}}([0,2p[){\Big )}+1}}
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
2
p
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
)
+
1
=
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
lim
p
∈
N
,
p
→
sup
(
N
)
p
)
+
1
{\displaystyle \displaystyle {={\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}2p\,\,{card}_{Q,{\cal {R}}}([0,1[){\Big )}+1=2\,\,{card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p{\Big )}+1}}
•(2)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
2
i
−
2
,
2
i
[
)
⨆
{
2
p
}
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
2
i
−
2
,
2
i
[
)
)
+
c
a
r
d
Q
,
R
(
{
2
p
}
)
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}\left(\left(\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[2i-2,2i[\right)\bigsqcup \{2p\}\right)=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([2i-2,2i[){\Big )}+{card}_{Q,{\cal {R}}}(\{2p\}){\bigg )}}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
2
i
−
2
,
2
i
[
)
)
+
1
)
=
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
2
i
−
2
,
2
i
[
)
)
+
1
=
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
0
,
2
[
)
)
+
1
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([2i-2,2i[){\Big )}+1{\bigg )}={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([2i-2,2i[){\Big )}+1={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,2[){\Big )}+1}}
=
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
)
+
1
=
2
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
)
+
1
{\displaystyle \displaystyle {={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}2\,\,{card}_{Q,{\cal {R}}}([0,1[){\Big )}+1=2\,\,{card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\Big )}+1}}
•[point où se rejoignent (1) et (2)]
=
2
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}
=
2
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {=2\,\,{card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}
Or
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
≠
2
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle {card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)+1\neq 2\,\,{card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
]
)
p
∈
N
]
)
≠
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
2
p
]
)
p
∈
N
]
)
{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}\neq {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
2
p
]
)
p
∈
N
]
)
=
2
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
]
)
p
∈
N
]
)
−
1
{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}-1}
On a bien
(
[
0
,
p
]
)
p
∈
N
≠
(
[
0
,
2
p
]
)
p
∈
N
{\displaystyle {([0,p])}_{p\in \mathbb {N} }\neq {([0,2p])}_{p\in \mathbb {N} }}
[Début point sensible]
f
∈
F
(
N
,
R
+
⨆
+
∞
)
,
s
t
r
i
c
t
.
↗
{\displaystyle f\in {\mathcal {F}}(\mathbb {N} ,\mathbb {R} _{+}\bigsqcup +\infty ),\,\,strict.\,\,\nearrow }
telle que
f
(
0
)
=
0
{\displaystyle f(0)=0}
et telle que
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
f
(
n
)
∈
+
∞
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}f(n)\in +\infty }}
lim
n
∈
N
,
n
→
sup
(
N
)
f
(
n
)
∈
+
∞
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow \sup(\mathbb {N} )}f(n)\in +\infty }}
qui est une expression qui est équivalente à l'expression "
lim
c
l
a
s
s
i
q
u
e
n
∈
N
,
n
→
sup
(
N
)
f
(
n
)
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \displaystyle {{\underset {n\in \mathbb {N} ,\,\,n\rightarrow \sup(\mathbb {N} )}{{\text{lim}}_{classique}}}f(n)=+\infty _{classique}}}
où
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle +\infty _{classique}}
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
f
(
n
)
=
f
(
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
n
)
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}f(n)=f(\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}n)}}
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
f
(
n
)
=
lim
n
∈
N
,
n
→
sup
(
N
)
f
(
n
)
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}f(n)=\lim _{n\in \mathbb {N} ,n\rightarrow \sup(\mathbb {N} )}f(n)}}
et "
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
n
=
lim
n
∈
N
,
n
→
sup
(
N
)
n
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}n=\lim _{n\in \mathbb {N} ,n\rightarrow \sup(\mathbb {N} )}n}}
(Cf. aussi "Définitions de
+
∞
{\displaystyle +\infty }
+
∞
″
{\displaystyle +\infty ''}
+
∞
f
{\displaystyle +\infty _{f}}
+
∞
F
(
R
)
{\displaystyle +\infty _{\cal {F(\mathbb {R} )}}}
R
′
{\displaystyle \mathbb {R} '}
R
″
{\displaystyle \mathbb {R} ''}
),
[Fin point sensible]
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
f
(
p
)
[
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
f
(
p
)
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
f
(
p
)
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,f(p)[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,f(p)[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,f(p)[)}}
•(1)
=
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)\,\,{card}_{Q,{\cal {R}}}([0,1[)={card}_{Q,{\cal {R}}}([0,1[)\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
f
(
i
−
1
)
,
f
(
i
)
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
f
(
i
−
1
)
,
f
(
i
)
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
0
,
f
(
i
)
−
f
(
i
−
1
)
[
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[f(i-1),f(i)[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([f(i-1),f(i)[)=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([0,f(i)-f(i-1)[)}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
(
f
(
i
)
−
f
(
i
−
1
)
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
(
f
(
i
)
−
f
(
i
−
1
)
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
lim
p
∈
N
,
p
→
sup
(
N
)
(
f
(
p
)
−
f
(
0
)
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}f(p)-f(0){\Big )}}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
lim
p
∈
N
,
p
→
sup
(
N
)
(
f
(
p
)
−
0
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}f(p)-0{\Big )}}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
f
(
lim
p
∈
N
,
p
→
sup
(
N
)
p
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f(\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p)}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
f
(
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f{\Bigg (}{{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {N} ,(\mathbb {N} \bigcap [0,p])_{p\in \mathbb {N} }{\bigg ]}}^{*}{\Bigg )}}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
f
(
c
a
r
d
Q
,
R
(
N
1
∗
)
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f{\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*}){\Big )}}}
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
f
(
p
)
]
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
f
(
p
)
]
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
f
(
p
)
]
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
f
(
p
)
[
⨆
{
f
(
p
)
}
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,f(p)])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,f(p)]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,f(p)])=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,f(p)[\bigsqcup \{f(p)\})}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
c
a
r
d
Q
,
R
(
[
0
,
f
(
p
)
[
)
+
c
a
r
d
Q
,
R
(
{
f
(
p
)
}
)
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}{card}_{Q,{\cal {R}}}([0,f(p)[)+{card}_{Q,{\cal {R}}}(\{f(p)\}){\Big )}}}
•(1)
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
c
a
r
d
Q
,
R
(
[
0
,
f
(
p
)
[
)
+
1
)
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
f
(
p
)
[
)
)
+
1
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\,\,{\Big (}{card}_{Q,{\cal {R}}}([0,f(p)[)+1{\Big )}={\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\,\,{card}_{Q,{\cal {R}}}([0,f(p)[){\Big )}+1}}
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
)
+
1
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
)
+
1
{\displaystyle \displaystyle {={\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)\,\,{card}_{Q,{\cal {R}}}([0,1[){\Big )}+1={card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p){\Big )}+1}}
•(2)
=
lim
p
∈
N
,
p
→
+
∞
(
c
a
r
d
Q
,
R
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
f
(
i
−
1
)
,
f
(
i
)
[
)
+
c
a
r
d
Q
,
R
(
{
f
(
p
)
}
)
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
f
(
i
−
1
)
,
f
(
i
)
[
)
)
+
1
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow +\infty }{\bigg (}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[f(i-1),f(i)[{\Big )}+{card}_{Q,{\cal {R}}}(\{f(p)\}){\bigg )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([f(i-1),f(i)[){\Big )}+1{\bigg )}}}
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
f
(
i
−
1
)
,
f
(
i
)
[
)
)
+
1
{\displaystyle \displaystyle {={\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([f(i-1),f(i)[){\bigg )}+1}}
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
0
,
f
(
i
)
−
f
(
i
−
1
)
[
)
)
+
1
{\displaystyle \displaystyle {={\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([0,f(i)-f(i-1)[){\bigg )}+1}}
=
(
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
(
f
(
i
)
−
f
(
i
−
1
)
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
)
+
1
{\displaystyle \displaystyle {={\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[){\bigg )}+1}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
(
f
(
i
)
−
f
(
i
−
1
)
)
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}{\bigg )}+1}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
lim
p
∈
N
,
p
→
sup
(
N
)
(
f
(
p
)
−
f
(
0
)
)
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}f(p)-f(0){\Big )}{\bigg )}+1}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
lim
p
∈
N
,
p
→
sup
(
N
)
(
f
(
p
)
−
0
)
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}f(p)-0{\Big )}{\bigg )}+1}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p){\bigg )}+1}}
•[point où se rejoignent (1) et (2)]
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
f
(
lim
p
∈
N
,
p
→
sup
(
N
)
p
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f(\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p)+1}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
f
(
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f{\Bigg (}{{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {N} ,(\mathbb {N} \bigcap [0,p])_{p\in \mathbb {N} }{\bigg ]}}^{*}{\Bigg )}+1}}
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
f
(
c
a
r
d
Q
,
R
(
N
1
∗
)
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f{\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*}){\Big )}+1}}
Fin du théorème
Remarque :
1) Soient
a
∈
R
⋃
{
−
sup
(
R
)
}
,
b
∈
R
,
a
<
b
{\displaystyle a\in \mathbb {R} \bigcup \{-\sup(\mathbb {R} )\},\,\,b\in \mathbb {R} ,\,\,a<b}
Soit
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle +\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}
Soit
f
∈
F
(
(
a
,
b
[
,
R
)
{\displaystyle \displaystyle {f\in {\mathcal {F}}((a,b[,\mathbb {R} )}}
telle que
lim
c
l
a
s
s
i
q
u
e
i
∈
(
a
,
b
[
,
i
→
b
−
f
(
i
)
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle {\underset {i\in (a,b[,i\rightarrow b^{-}}{{\text{lim}}_{classique}}}f(i)=+\infty _{classique}}
Alors on pose :
lim
i
∈
(
a
,
b
[
,
i
→
b
−
f
(
i
)
=
sup
(
+
∞
)
{\displaystyle \lim _{i\in (a,b[,i\rightarrow b^{-}}f(i)=\sup(+\infty )}
lim
p
∈
N
,
p
→
sup
(
N
)
p
=
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
1
=
∑
i
∈
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
)
1
=
∑
i
∈
(
lim
p
∈
N
,
p
→
sup
(
N
)
(
N
⋂
[
0
,
p
]
)
)
∖
{
0
}
1
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}1=\sum _{\displaystyle {i\in \lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}}}1=\sum _{\displaystyle {i\in {\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}(\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}}}1=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1}}
=
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
=
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
1
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}={card}_{Q,{\cal {R}}}(N_{1}^{*})=\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1})}}
ou dit autrement :
lim
p
∈
N
,
p
→
sup
(
N
)
p
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
)
)
=
c
a
r
d
Q
,
R
(
(
lim
p
∈
N
,
p
→
sup
(
N
)
(
N
⋂
[
0
,
p
]
)
)
∖
{
0
}
)
{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}={card}_{Q,{\cal {R}}}{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}{\bigg )}={card}_{Q,{\cal {R}}}{\bigg (}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}(\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}{\bigg )}}}
=
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
=
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
1
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}={card}_{Q,{\cal {R}}}(N_{1}^{*})=\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1})}}
[Début point sensible]
Soit
f
∈
F
(
N
,
R
+
⨆
+
∞
)
,
s
t
r
i
c
t
.
↗
{\displaystyle f\in {\mathcal {F}}(\mathbb {N} ,\mathbb {R} _{+}\bigsqcup +\infty ),\,\,strict.\,\,\nearrow }
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
f
(
n
)
∈
+
∞
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}f(n)\in +\infty }}
lim
n
∈
N
,
n
→
sup
(
N
)
f
(
n
)
∈
+
∞
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,\,\,n\rightarrow \sup(\mathbb {N} )}f(n)\in +\infty }}
qui est une expression qui est équivalente à l'expression "
lim
c
l
a
s
s
i
q
u
e
n
∈
N
,
n
→
sup
(
N
)
f
(
n
)
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \displaystyle {{\underset {n\in \mathbb {N} ,\,\,n\rightarrow \sup(\mathbb {N} )}{{\text{lim}}_{classique}}}f(n)=+\infty _{classique}}}
où
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle +\infty _{classique}}
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
f
(
n
)
=
f
(
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
n
)
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}f(n)=f(\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}n)}}
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
f
(
n
)
=
lim
n
∈
N
,
n
→
sup
(
N
)
f
(
n
)
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}f(n)=\lim _{n\in \mathbb {N} ,n\rightarrow \sup(\mathbb {N} )}f(n)}}
et "
lim
n
∈
N
,
n
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
n
=
lim
n
∈
N
,
n
→
sup
(
N
)
n
{\displaystyle \displaystyle {\lim _{n\in \mathbb {N} ,n\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}n=\lim _{n\in \mathbb {N} ,n\rightarrow \sup(\mathbb {N} )}n}}
[Fin point sensible]
Alors :
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
=
f
(
lim
p
∈
N
,
p
→
sup
(
N
)
p
)
=
f
(
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
(
N
⋂
[
0
,
p
]
)
)
∖
{
0
}
1
)
=
f
(
∑
i
∈
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
)
1
)
{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)=f(\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p)=f{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in {\Big (}(\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}}1{\bigg )}=f{\bigg (}\sum _{\displaystyle {i\in \lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}}}1{\bigg )}}}
=
f
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
)
=
f
(
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
)
=
f
(
c
a
r
d
Q
,
R
(
N
1
∗
)
)
=
f
(
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
1
)
)
{\displaystyle \displaystyle {=f{\bigg (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\bigg )}=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}{\Bigg )}=f{\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*}){\Big )}=f{\Big (}\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1}){\Big )}}}
ou dit autrement :
lim
p
∈
N
,
p
→
sup
(
N
)
f
(
p
)
=
f
(
lim
p
∈
N
,
p
→
sup
(
N
)
p
)
=
f
(
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
)
)
=
f
(
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
)
)
)
{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)=f(\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p)=f{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}{\bigg )}=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}{\bigg )}{\Bigg )}}}
=
f
(
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
(
N
⋂
[
0
,
p
]
)
)
∖
{
0
}
)
)
=
f
(
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
)
=
f
(
c
a
r
d
Q
,
R
(
N
1
∗
)
)
=
f
(
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
1
)
)
{\displaystyle \displaystyle {=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}{\bigg )}{\Bigg )}=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}{\Bigg )}=f{\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*}){\Big )}=f{\Big (}\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1}){\Big )}}}
Propositions concernant certains intervalles
I
{\displaystyle I}
R
{\displaystyle \mathbb {R} }
P
V
2
(
R
)
{\displaystyle {PV2}(\mathbb {R} )}
[ modifier | modifier le wikicode ]
R
+
{\displaystyle \mathbb {R} _{+}}
[ modifier | modifier le wikicode ] Début d’un théorème
Remarque : J'hésite, ici, à omettre la notation "
~
{\displaystyle {\widetilde {\,\,\,\,\,}}}
v
o
l
1
~
{\displaystyle {\widetilde {{vol}^{1}}}}
sup
(
N
)
=
sup
(
R
)
=
+
∞
N
=
+
∞
R
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \sup(\mathbb {N} )=\sup(\mathbb {R} )=+\infty _{\mathbb {N} }=+\infty _{\mathbb {R} }=+\infty _{classique}}
R
{\displaystyle {\mathcal {R}}}
R
{\displaystyle \mathbb {R} }
O
{\displaystyle O}
R
1
,
+
=
[
R
+
,
(
[
0
,
r
]
)
r
∈
N
]
{\displaystyle R_{1,+}={\Big [}\mathbb {R} _{+},{([0,r])}_{r\in \mathbb {N} }{\Big ]}}
R
2
,
+
=
[
R
+
,
(
[
0
,
r
[
)
r
∈
N
]
{\displaystyle R_{2,+}={\Big [}\mathbb {R} _{+},{([0,r[)}_{r\in \mathbb {N} }{\Big ]}}
N
1
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
{\displaystyle \displaystyle {N_{1}={\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}}
N
1
∗
=
N
1
∖
{
0
}
{\displaystyle \displaystyle {N_{1}^{*}=N_{1}\setminus \{0\}}}
c
a
r
d
Q
,
R
(
R
2
,
+
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
(
c
a
r
d
Q
,
R
(
N
1
)
−
1
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
v
o
l
1
~
(
R
2
,
+
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{2,+})={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)={\Big (}{card}_{Q,{\cal {R}}}(N_{1})-1{\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[)={\widetilde {{vol}^{1}}}(R_{2,+})\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
et
c
a
r
d
Q
,
R
(
R
1
,
+
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
=
(
c
a
r
d
Q
,
R
(
N
1
)
−
1
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
=
v
o
l
1
~
(
R
1
,
+
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{1,+})={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)+1={\Big (}{card}_{Q,{\cal {R}}}(N_{1})-1{\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[)+1={\widetilde {{vol}^{1}}}(R_{1,+})\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}
R
2
,
+
{\displaystyle R_{2,+}}
R
+
{\displaystyle \mathbb {R} _{+}}
Fin du théorème
Démonstration :
On a :
c
a
r
d
Q
,
R
(
R
1
,
+
)
=
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
]
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
p
]
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
p
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{1,+})={card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,p]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p])}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
i
−
1
,
i
[
)
⨆
{
p
}
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
)
+
c
a
r
d
Q
,
R
(
{
p
}
)
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}\left(\left(\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[i-1,i[\right)\bigsqcup \{p\}\right)=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+{card}_{Q,{\cal {R}}}(\{p\}){\bigg )}}}
lim
p
∈
N
,
p
→
sup
(
N
)
(
(
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
)
+
1
)
=
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
)
+
1
{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1{\bigg )}={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1}}
=
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
)
+
1
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
(
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
)
+
1
=
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[){\Big )}+1={card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\Big )}+1={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
+
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}
c
a
r
d
Q
,
R
(
R
2
,
+
)
=
c
a
r
d
Q
,
R
(
[
R
+
,
(
[
0
,
p
[
)
p
∈
N
]
)
=
c
a
r
d
Q
,
R
(
lim
p
∈
N
,
p
→
sup
(
N
)
[
0
,
p
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
[
0
,
p
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{2,+})={card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,p[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p[)}}
=
lim
p
∈
N
,
p
→
sup
(
N
)
c
a
r
d
Q
,
R
(
⨆
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
[
i
−
1
,
i
[
)
=
lim
p
∈
N
,
p
→
sup
(
N
)
∑
i
∈
(
N
⋂
[
0
,
p
]
)
∖
{
0
}
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
i
−
1
,
i
[
)
{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[i-1,i[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[)=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([i-1,i[)}}
=
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
=
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
∑
i
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
1
=
c
a
r
d
Q
,
R
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[)={card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}}
Remarque :
Soit
R
{\displaystyle {\mathcal {R}}}
R
{\displaystyle \mathbb {R} }
O
{\displaystyle O}
De plus, soit
p
∈
N
{\displaystyle p\in \mathbb {N} }
Si
p
→
c
l
a
s
s
i
q
u
e
sup
(
N
)
où
sup
(
N
)
=
+
∞
c
l
a
s
s
i
q
u
e
et où
+
∞
c
l
a
s
s
i
q
u
e
est considéré comme un point
{\displaystyle \displaystyle {{\mbox{Si}}\,\,p{\underset {classique}{\rightarrow }}\sup(\mathbb {N} )\,\,{\mbox{où}}}\,\,\sup(\mathbb {N} )=+\infty _{classique}\,\,{\mbox{et où}}\,\,+\infty _{classique}\,\,{\mbox{est considéré comme un point}}}
alors
p
−
1
→
c
l
a
s
s
i
q
u
e
sup
(
N
)
{\displaystyle p-1{\underset {classique}{\rightarrow }}\sup(\mathbb {N} )}
lim
c
l
a
s
s
i
q
u
e
p
∈
N
,
p
→
sup
(
N
)
(
p
−
1
)
=
lim
c
l
a
s
s
i
q
u
e
p
∈
N
,
p
→
sup
(
N
)
p
=
sup
(
N
)
{\displaystyle \displaystyle {{\underset {p\in \mathbb {N} ,\,\,p\to \sup(\mathbb {N} )}{{\text{lim}}_{classique}}}(p-1)={\underset {p\in \mathbb {N} ,\,\,p\to \sup(\mathbb {N} )}{{\text{lim}}_{classique}}}p=\sup(\mathbb {N} )}}
⋯
{\displaystyle \cdots }
et
c
a
r
d
Q
,
R
(
N
⋂
[
0
,
p
]
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(\mathbb {N} \bigcap [0,p])}}
=
c
a
r
d
Q
,
R
(
[
[
0
,
p
]
]
}
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,p]\!]\})}}
=
p
+
1
{\displaystyle \displaystyle {=p+1}}
et
c
a
r
d
Q
,
R
(
{
[
[
0
,
lim
c
l
a
s
s
i
q
u
e
p
∈
N
,
p
→
sup
(
N
)
p
]
]
}
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(\{[\![0,{\underset {p\in \mathbb {N} ,\,\,p\to \sup(\mathbb {N} )}{{\text{lim}}_{classique}}}p]\!]\})}}
c
a
r
d
Q
,
R
(
{
[
[
0
,
sup
(
N
)
]
]
}
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(\{[\![0,\sup(\mathbb {N} )]\!]\})}}
=
c
a
r
d
Q
,
R
(
N
⨆
{
sup
(
N
)
}
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}(\mathbb {N} \bigsqcup \{\sup(\mathbb {N} )\})}}
(
=
c
a
r
d
Q
,
R
(
N
)
+
1
)
{\displaystyle {\Big (}={card}_{Q,{\mathcal {R}}}(\mathbb {N} )+1{\Big )}}
=
c
a
r
d
Q
,
R
(
N
¯
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}({\overline {\mathbb {N} }})}}
Si
p
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
∈
+
∞
où
+
∞
est considéré comme un ensemble tel que
{\displaystyle \displaystyle {{\mbox{Si}}\,\,p\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )\in +\infty \,\,{\mbox{où}}\,\,+\infty \,\,{\mbox{est considéré comme un ensemble tel que}}}}
+
∞
=
{
x
|
∀
a
∈
R
,
x
>
a
}
{\displaystyle \displaystyle {+\infty =\{x\,\,|\,\,\forall a\in \mathbb {R} ,\,\,x>a\}}}
alors
p
−
1
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
−
1
∈
+
∞
{\displaystyle p-1\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )-1\in +\infty }
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
−
1
≠
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
{\displaystyle \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )-1\neq \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}
⋯
{\displaystyle \cdots }
et
c
a
r
d
Q
,
R
(
[
[
0
,
lim
c
l
a
s
s
i
q
u
e
p
∈
N
⨆
{
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
}
,
p
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
p
]
]
)
{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([\![0,\displaystyle {{\underset {p\in \mathbb {N} \bigsqcup \{\sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )\},\,\,p\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}{{\text{lim}}_{classique}}}p}]\!])}}
=
c
a
r
d
Q
,
R
(
[
[
0
,
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
]
]
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,\sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )]\!])}}
et
=
c
a
r
d
Q
,
R
(
[
[
0
,
lim
p
∈
N
,
p
→
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
p
]
]
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,{\displaystyle {\lim _{p\in \mathbb {N} ,\,\,p\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}p}]\!])}}}
=
c
a
r
d
Q
,
R
(
[
[
0
,
lim
p
∈
N
,
p
→
sup
(
N
)
p
]
]
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,{\displaystyle {\lim _{p\in \mathbb {N} ,\,\,p\rightarrow \sup(\mathbb {N} )}p}]\!])}}}
=
c
a
r
d
Q
,
R
(
[
[
0
,
sup
n
o
n
c
l
a
s
s
i
q
u
e
,
R
(
N
)
]
]
)
{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,\sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )]\!])}}
Début d’un théorème
Ici,
sup
(
N
)
=
sup
(
R
)
=
+
∞
N
=
+
∞
R
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \sup(\mathbb {N} )=\sup(\mathbb {R} )=+\infty _{\mathbb {N} }=+\infty _{\mathbb {R} }=+\infty _{classique}}
R
{\displaystyle {\mathcal {R}}}
R
{\displaystyle \mathbb {R} }
O
{\displaystyle O}
R
2
=
[
R
,
(
]
−
r
,
r
[
)
r
∈
N
]
{\displaystyle R_{2}={\Big [}\mathbb {R} ,{(]-r,r[)}_{r\in \mathbb {N} }{\Big ]}}
R
2
,
+
=
[
R
+
,
(
[
0
,
r
[
)
r
∈
N
]
{\displaystyle R_{2,+}={\Big [}\mathbb {R} _{+},{([0,r[)}_{r\in \mathbb {N} }{\Big ]}}
R
2
,
−
=
[
R
−
,
(
]
−
r
,
0
]
)
r
∈
N
]
{\displaystyle R_{2,-}={\Big [}\mathbb {R} _{-},{(]-r,0])}_{r\in \mathbb {N} }{\Big ]}}
R
2
,
+
∗
=
R
2
,
+
∖
{
0
}
{\displaystyle R_{2,+}^{*}=R_{2,+}\setminus \{0\}}
R
2
,
−
∗
=
R
2
,
−
∖
{
0
}
{\displaystyle R_{2,-}^{*}=R_{2,-}\setminus \{0\}}
N
1
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
{\displaystyle \displaystyle {N_{1}={\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}}
N
1
∗
=
N
1
∖
{
0
}
{\displaystyle \displaystyle {N_{1}^{*}=N_{1}\setminus \{0\}}}
−
N
1
=
[
−
N
,
(
−
N
⋂
[
−
p
,
0
]
)
p
∈
N
]
{\displaystyle \displaystyle {-N_{1}={\Big [}-\mathbb {N} ,{(-\mathbb {N} \bigcap [-p,0])}_{p\in \mathbb {N} }{\Big ]}}}
−
N
1
∗
=
−
N
1
∖
{
0
}
{\displaystyle \displaystyle {-N_{1}^{*}=-N_{1}\setminus \{0\}}}
Z
1
=
[
Z
,
(
Z
⋂
[
−
p
,
p
]
)
p
∈
N
]
{\displaystyle \displaystyle {Z_{1}={\Big [}\mathbb {Z} ,{(\mathbb {Z} \bigcap [-p,p])}_{p\in \mathbb {N} }{\Big ]}}}
Z
1
∗
=
Z
1
∖
{
0
}
{\displaystyle \displaystyle {Z_{1}^{*}=Z_{1}\setminus \{0\}}}
c
a
r
d
Q
,
R
(
R
2
,
−
∗
)
=
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2,-}^{*})={card}_{Q,{\cal {R}}}(R_{2,+}^{*})}
R
2
=
R
2
,
−
∗
⨆
{
0
}
⨆
R
2
,
+
∗
{\displaystyle \displaystyle {R_{2}=R_{2,-}^{*}\bigsqcup \{0\}\bigsqcup R_{2,+}^{*}}}
c
a
r
d
Q
,
R
(
R
2
)
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2})}
=
c
a
r
d
Q
,
R
(
R
2
,
−
∗
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
+
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
{\displaystyle ={card}_{Q,{\cal {R}}}(R_{2,-}^{*})+{card}_{Q,{\cal {R}}}(\{0\})+{card}_{Q,{\cal {R}}}(R_{2,+}^{*})}
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}(R_{2,+}^{*})+{card}_{Q,{\cal {R}}}(\{0\})}
[c'est-à-dire
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
+
1
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}(R_{2,+}^{*})+1}
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
∖
{
0
}
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus \{0\})+{card}_{Q,{\cal {R}}}(\{0\})}
=
2
(
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
c
a
r
d
Q
,
R
(
{
0
}
)
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
{\displaystyle =2\,\,{\Big (}{card}_{Q,{\cal {R}}}(R_{2,+})-{card}_{Q,{\cal {R}}}(\{0\}){\Big )}+{card}_{Q,{\cal {R}}}(\{0\})}
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
c
a
r
d
Q
,
R
(
{
0
}
)
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}(R_{2,+})-{card}_{Q,{\cal {R}}}(\{0\})}
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
1
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}(R_{2,+})-1}
c
a
r
d
Q
,
R
(
R
2
)
=
c
a
r
d
Q
,
R
(
R
2
,
−
∗
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
+
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2})={card}_{Q,{\cal {R}}}(R_{2,-}^{*})+{card}_{Q,{\cal {R}}}(\{0\})+{card}_{Q,{\cal {R}}}(R_{2,+}^{*})}
c
a
r
d
Q
,
R
(
R
2
,
−
∗
)
=
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2,-}^{*})={card}_{Q,{\cal {R}}}(R_{2,+}^{*})}
c
a
r
d
Q
,
R
(
R
2
,
+
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2,+})={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)}
c
a
r
d
Q
,
R
(
−
N
1
∗
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
{\displaystyle {card}_{Q,{\cal {R}}}(-N_{1}^{*})={card}_{Q,{\cal {R}}}(N_{1}^{*})}
c
a
r
d
Q
,
R
(
Z
1
∗
)
=
c
a
r
d
Q
,
R
(
−
N
1
∗
⨆
N
1
∗
)
=
c
a
r
d
Q
,
R
(
−
N
1
∗
)
+
c
a
r
d
Q
,
R
(
N
1
∗
)
=
c
a
r
d
Q
,
R
(
N
1
∗
)
+
c
a
r
d
Q
,
R
(
N
1
∗
)
=
2
c
a
r
d
Q
,
R
(
N
1
∗
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(Z_{1}^{*})={card}_{Q,{\cal {R}}}(-N_{1}^{*}\bigsqcup N_{1}^{*})={card}_{Q,{\cal {R}}}(-N_{1}^{*})+{card}_{Q,{\cal {R}}}(N_{1}^{*})={card}_{Q,{\cal {R}}}(N_{1}^{*})+{card}_{Q,{\cal {R}}}(N_{1}^{*})=2\,\,{card}_{Q,{\cal {R}}}(N_{1}^{*})}}
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
=
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
1
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
−
1
=
c
a
r
d
Q
,
R
(
N
1
∗
)
(
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
)
−
1
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2,+}^{*})={card}_{Q,{\cal {R}}}(R_{2,+})-1={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-1={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,1[)+1{\Big )}-1}
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
+
1
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
=
c
a
r
d
Q
,
R
(
N
1
∗
)
=
c
a
r
d
Q
,
R
(
N
1
)
−
1
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(R_{2,+}^{*})+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(N_{1}^{*})={card}_{Q,{\cal {R}}}(N_{1})-1}}
c
a
r
d
Q
,
R
(
R
2
)
=
c
a
r
d
Q
,
R
(
R
2
,
−
∗
)
+
c
a
r
d
Q
,
R
(
{
0
}
)
+
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
∗
)
+
1
=
2
(
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
1
)
+
1
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
1
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2})={card}_{Q,{\cal {R}}}(R_{2,-}^{*})+{card}_{Q,{\cal {R}}}(\{0\})+{card}_{Q,{\cal {R}}}(R_{2,+}^{*})=2\,\,{card}_{Q,{\cal {R}}}(R_{2,+}^{*})+1=2\,\,{\Big (}{card}_{Q,{\cal {R}}}(R_{2,+})-1{\Big )}+1=2\,\,{card}_{Q,{\cal {R}}}(R_{2,+})-1}
=
2
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
−
1
=
c
a
r
d
Q
,
R
(
Z
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
−
1
=
c
a
r
d
Q
,
R
(
Z
1
∗
)
(
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
)
−
1
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-1={card}_{Q,{\cal {R}}}(Z_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-1={card}_{Q,{\cal {R}}}(Z_{1}^{*})\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,1[)+1{\Big )}-1}
c
a
r
d
Q
,
R
(
R
2
)
+
1
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
=
c
a
r
d
Q
,
R
(
Z
1
∗
)
=
c
a
r
d
Q
,
R
(
Z
1
)
−
1
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(R_{2})+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(Z_{1}^{*})={card}_{Q,{\cal {R}}}(Z_{1})-1}}
Fin du théorème
Début d’un théorème
Ici,
sup
(
N
)
=
sup
(
R
)
=
+
∞
N
=
+
∞
R
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \sup(\mathbb {N} )=\sup(\mathbb {R} )=+\infty _{\mathbb {N} }=+\infty _{\mathbb {R} }=+\infty _{classique}}
R
{\displaystyle {\mathcal {R}}}
R
{\displaystyle \mathbb {R} }
O
{\displaystyle O}
On pose :
R
2
,
+
=
[
R
+
,
(
[
0
,
r
[
)
r
∈
N
]
{\displaystyle {R}_{2,+}={\Big [}{\mathbb {R} }_{+},{([0,r[)}_{r\in \mathbb {N} }{\Big ]}}
N
1
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
{\displaystyle N_{1}={\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}
a
,
b
∈
R
+
:
a
≤
b
{\displaystyle a,b\in \mathbb {R} _{+}\,\,:\,\,a\leq b}
c
a
r
d
Q
,
R
(
R
2
,
+
∖
[
0
,
a
]
)
=
c
a
r
d
Q
,
R
(
]
a
,
b
]
)
+
c
a
r
d
Q
,
R
(
R
2
,
+
∖
[
0
,
b
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])={card}_{Q,{\cal {R}}}(]a,b])+{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,b])}}
c
a
r
d
Q
,
R
(
R
2
)
{\displaystyle {card}_{Q,{\cal {R}}}(R_{2})}
=
2
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
1
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}({R}_{2,+})-1}
=
2
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
−
1
{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}({N_{1}}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-1}
=
2
(
c
a
r
d
Q
,
R
(
N
1
)
−
1
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
−
1
{\displaystyle =2\,\,{\Big (}{card}_{Q,{\cal {R}}}(N_{1})-1{\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[)-1}
Fin du théorème
Début d’un théorème
Ici,
sup
(
N
)
=
sup
(
R
)
=
+
∞
N
=
+
∞
R
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \sup(\mathbb {N} )=\sup(\mathbb {R} )=+\infty _{\mathbb {N} }=+\infty _{\mathbb {R} }=+\infty _{classique}}
R
{\displaystyle {\mathcal {R}}}
R
{\displaystyle \mathbb {R} }
O
{\displaystyle O}
On pose :
R
2
,
+
=
[
R
+
,
(
[
0
,
r
[
)
r
∈
N
]
{\displaystyle {R}_{2,+}={\Big [}{\mathbb {R} }_{+},{([0,r[)}_{r\in \mathbb {N} }{\Big ]}}
N
1
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
{\displaystyle N_{1}={\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}
a
∈
R
+
{\displaystyle a\in \mathbb {R} _{+}}
c
a
r
d
Q
,
R
(
R
2
,
+
∖
[
0
,
a
]
)
+
1
{\displaystyle {card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])+1}
=
c
a
r
d
Q
,
R
(
R
2
,
+
∖
[
0
,
a
[
)
{\displaystyle ={card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a[)}
=
c
a
r
d
Q
,
R
(
R
2
,
+
)
−
c
a
r
d
Q
,
R
(
[
0
,
a
[
)
{\displaystyle ={card}_{Q,{\cal {R}}}(R_{2,+})-{card}_{Q,{\cal {R}}}([0,a[)}
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
−
a
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle ={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-a\,\,{card}_{Q,{\cal {R}}}([0,1[)}
=
(
c
a
r
d
Q
,
R
(
N
1
∗
)
−
a
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle ={\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*})-a{\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}
=
(
c
a
r
d
Q
,
R
(
N
1
∗
)
−
a
)
(
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
)
{\displaystyle ={\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*})-a{\Big )}\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,1[)+1{\Big )}}
c
a
r
d
Q
,
R
(
R
2
,
+
∖
[
0
,
a
]
)
+
1
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
=
c
a
r
d
Q
,
R
(
N
1
∗
)
−
a
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(N_{1}^{*})-a}}
Soit
a
∈
R
−
{\displaystyle a\in \mathbb {R} _{-}}
c
a
r
d
Q
,
R
(
R
2
,
+
⋃
]
a
,
0
[
)
+
1
{\displaystyle {card}_{Q,{\cal {R}}}({R}_{2,+}\bigcup ]a,0[)+1}
=
c
a
r
d
Q
,
R
(
R
2
,
+
⋃
[
a
,
0
[
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}({R}_{2,+}\bigcup [a,0[)}}
=
c
a
r
d
Q
,
R
(
R
2
,
+
)
+
c
a
r
d
Q
,
R
(
[
a
,
0
[
)
{\displaystyle ={card}_{Q,{\cal {R}}}(R_{2,+})+{card}_{Q,{\cal {R}}}([a,0[)}
=
c
a
r
d
Q
,
R
(
N
1
∗
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
−
a
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle ={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-a\,\,{card}_{Q,{\cal {R}}}([0,1[)}
=
(
c
a
r
d
Q
,
R
(
N
1
∗
)
−
a
)
c
a
r
d
Q
,
R
(
[
0
,
1
[
)
{\displaystyle ={\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*})-a{\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}
=
(
c
a
r
d
Q
,
R
(
N
1
∗
)
−
a
)
(
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
)
{\displaystyle ={\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*})-a{\Big )}\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,1[)+1{\Big )}}
c
a
r
d
Q
,
R
(
R
2
,
+
⋃
]
a
,
0
[
)
+
1
c
a
r
d
Q
,
R
(
]
0
,
1
[
)
+
1
=
c
a
r
d
Q
,
R
(
N
1
∗
)
−
a
{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({R}_{2,+}\bigcup ]a,0[)+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(N_{1}^{*})-a}}
Soit
a
∈
R
+
{\displaystyle a\in \mathbb {R} _{+}}
c
a
r
d
Q
,
R
(
R
2
,
+
∖
[
0
,
a
]
)
=
c
a
r
d
Q
,
R
(
R
2
,
−
∖
[
−
a
,
0
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])={card}_{Q,{\cal {R}}}({R}_{2,-}\setminus [-a,0])}}
c
a
r
d
Q
,
R
(
R
2
,
+
∖
[
0
,
a
[
)
=
c
a
r
d
Q
,
R
(
R
2
,
−
∖
]
−
a
,
0
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a[)={card}_{Q,{\cal {R}}}({R}_{2,-}\setminus ]-a,0])}}
Fin du théorème
R
2
{\displaystyle \mathbb {R} ^{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
O
2
(
0
,
0
)
{\displaystyle O_{2}(0,0)}
R
2
{\displaystyle {\cal {R}}_{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
[ modifier | modifier le wikicode ] Début d’un théorème
Ici,
sup
(
N
)
=
sup
(
R
)
=
+
∞
N
=
+
∞
R
=
+
∞
c
l
a
s
s
i
q
u
e
{\displaystyle \sup(\mathbb {N} )=\sup(\mathbb {R} )=+\infty _{\mathbb {N} }=+\infty _{\mathbb {R} }=+\infty _{classique}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
∀
i
∈
N
n
∗
,
R
i
{\displaystyle \forall i\in \mathbb {N} _{n}^{*},\,\,{\cal {R}}_{i}}
R
i
{\displaystyle \mathbb {R} ^{i}}
O
i
(
0
)
j
∈
N
i
∗
{\displaystyle O_{i}{(0)}_{j\in \mathbb {N} _{i}^{*}}}
1) Suivant un plafonnement carré, autour de l'origine, suivant les 2 axes orthonormés
(
O
2
x
)
{\displaystyle (O_{2}x)}
(
O
2
y
)
{\displaystyle (O_{2}y)}
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}}}
Ici, on considère que :
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
=
lim
r
∈
N
,
r
→
sup
(
N
)
[
−
r
,
r
]
{\displaystyle \displaystyle {{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}[-r,r]}}
et on a :
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
=
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
2
{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}={{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}}^{2}}}
c
a
r
d
Q
,
R
2
(
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
)
=
c
a
r
d
Q
,
R
2
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
2
)
=
(
c
a
r
d
Q
,
R
2
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
)
2
=
c
a
r
d
Q
,
R
1
2
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}={{\Bigg (}{card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}{\Bigg )}}^{2}={card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}
2) Suivant un plafonnement sphérique, autour de l'origine, noté
[
R
2
,
(
B
R
2
(
O
2
,
r
)
¯
)
r
∈
N
]
{\displaystyle \displaystyle {{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}}}
Ici, on considère que :
[
R
2
,
(
B
R
2
(
O
2
,
r
)
¯
)
r
∈
N
]
=
lim
r
∈
N
,
r
→
+
∞
B
R
2
(
O
2
,
r
)
¯
{\displaystyle \displaystyle {{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}=\lim _{r\in \mathbb {N} ,r\rightarrow +\infty }{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}}}
∀
r
∈
N
,
B
R
2
(
O
2
,
r
)
¯
{\displaystyle \forall r\in \mathbb {N} ,\,\,{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}}
R
2
{\displaystyle \mathbb {R} ^{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
lim
r
∈
N
,
r
→
sup
(
N
)
B
R
2
(
O
2
,
r
)
¯
=
[
R
2
,
(
B
R
2
(
O
2
,
r
)
¯
)
r
∈
N
]
{\displaystyle \displaystyle {\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}={\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}}}
c
a
r
d
Q
,
R
2
(
[
R
2
,
(
B
R
2
(
O
2
,
r
)
¯
)
r
∈
N
]
)
=
c
a
r
d
Q
,
R
2
(
lim
r
∈
N
,
r
→
sup
(
N
)
B
R
2
(
O
2
,
r
)
¯
)
=
lim
r
∈
N
,
r
→
sup
(
N
)
c
a
r
d
Q
,
R
2
(
B
R
2
(
O
2
,
r
)
¯
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\Bigg (}{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}{\Bigg )}={card}_{Q,{\cal {R}}_{2}}{\Big (}\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}}
∀
r
∈
N
,
c
a
r
d
Q
,
R
2
(
B
R
2
(
O
2
,
r
)
¯
)
=
?
{\displaystyle \forall r\in \mathbb {N} ,\,\,{card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}=?}
c
a
r
d
Q
,
R
2
(
B
R
2
(
O
2
,
1
)
¯
)
=
π
c
a
r
d
Q
,
R
1
2
(
[
0
,
1
[
)
+
π
c
a
r
d
Q
,
R
1
(
[
0
,
1
[
)
+
1
{\displaystyle {card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},1)}}{\Big )}=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}([0,1[)+\pi \,\,{card}_{Q,{\cal {R}}_{1}}([0,1[)+1}
c
a
r
d
Q
,
R
1
(
[
0
,
1
[
)
=
c
a
r
d
Q
,
R
1
(
[
0
,
1
]
)
−
1
{\displaystyle {card}_{Q,{\cal {R}}_{1}}([0,1[)={card}_{Q,{\cal {R}}_{1}}([0,1])-1}
c
a
r
d
Q
,
R
2
(
[
0
,
1
[
2
)
=
c
a
r
d
Q
,
R
1
2
(
[
0
,
1
[
)
=
(
c
a
r
d
Q
,
R
1
(
[
0
,
1
]
)
−
1
)
2
=
c
a
r
d
Q
,
R
1
2
(
[
0
,
1
]
)
−
2
c
a
r
d
Q
,
R
1
(
[
0
,
1
]
)
+
1
{\displaystyle {card}_{Q,{\cal {R}}_{2}}({[0,1[}^{2})={card}_{Q,{\cal {R}}_{1}}^{2}([0,1[)={{\Big (}{card}_{Q,{\cal {R}}_{1}}([0,1])-1{\Big )}}^{2}={card}_{Q,{\cal {R}}_{1}}^{2}([0,1])-2\,\,{card}_{Q,{\cal {R}}_{1}}([0,1])+1}
c
a
r
d
Q
,
R
2
(
B
R
2
(
O
2
,
1
)
¯
)
=
π
c
a
r
d
Q
,
R
1
2
(
[
0
,
1
]
)
−
π
c
a
r
d
Q
,
R
1
(
[
0
,
1
]
)
+
1
{\displaystyle {card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},1)}}{\Big )}=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}([0,1])-\pi \,\,{card}_{Q,{\cal {R}}_{1}}([0,1])+1}
∀
r
∈
N
,
c
a
r
d
Q
,
R
2
(
B
R
2
(
O
2
,
r
)
¯
)
=
π
c
a
r
d
Q
,
R
1
2
(
[
0
,
r
]
)
−
π
c
a
r
d
Q
,
R
1
(
[
0
,
r
]
)
+
1
{\displaystyle \forall r\in \mathbb {N} ,\,\,{card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}([0,r])-\pi \,\,{card}_{Q,{\cal {R}}_{1}}([0,r])+1}
c
a
r
d
Q
,
R
2
(
[
R
2
,
(
B
R
2
(
O
2
,
r
)
¯
)
r
∈
N
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\Bigg (}{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}{\Bigg )}}}
=
c
a
r
d
Q
,
R
2
(
lim
r
∈
N
,
r
→
sup
(
N
)
B
R
2
(
O
2
,
r
)
¯
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}{\Big (}\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}}
=
lim
r
∈
N
,
r
→
sup
(
N
)
c
a
r
d
Q
,
R
2
(
B
R
2
(
O
2
,
r
)
¯
)
{\displaystyle \displaystyle {=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}}
=
lim
r
∈
N
,
r
→
sup
(
N
)
(
π
c
a
r
d
Q
,
R
1
2
(
[
0
,
r
]
)
−
π
c
a
r
d
Q
,
R
1
(
[
0
,
r
]
)
+
1
)
{\displaystyle \displaystyle {=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\Big (}\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}([0,r])-\pi \,\,{card}_{Q,{\cal {R}}_{1}}([0,r])+1{\Big )}}}
=
π
lim
r
∈
N
,
r
→
sup
(
N
)
c
a
r
d
Q
,
R
1
2
(
[
0
,
r
]
)
−
π
lim
r
∈
R
+
,
r
→
sup
(
N
)
c
a
r
d
Q
,
R
1
(
[
0
,
r
]
)
+
1
{\displaystyle \displaystyle {=\pi \,\,\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}_{1}}^{2}([0,r])-\pi \,\,\lim _{r\in \mathbb {R} _{+},r\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}_{1}}([0,r])+1}}
=
π
c
a
r
d
Q
,
R
1
2
(
lim
r
∈
N
,
r
→
sup
(
N
)
[
0
,
r
]
)
−
π
c
a
r
d
Q
,
R
1
(
lim
r
∈
N
,
r
→
sup
(
N
)
[
0
,
r
]
)
+
1
{\displaystyle \displaystyle {=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}{\Big (}\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}[0,r]{\Big )}-\pi \,\,{card}_{Q,{\cal {R}}_{1}}{\Big (}\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}[0,r]{\Big )}+1}}
=
π
c
a
r
d
Q
,
R
1
2
(
[
R
+
,
(
[
0
,
r
]
)
r
∈
N
]
)
−
π
c
a
r
d
Q
,
R
1
(
[
R
+
,
(
[
0
,
r
]
)
r
∈
N
]
)
+
1
{\displaystyle \displaystyle {=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}-\pi \,\,{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}+1}}
=
1
4
π
(
c
a
r
d
Q
,
R
1
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
+
1
)
2
−
1
2
π
(
c
a
r
d
Q
,
R
1
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
+
1
)
+
1
{\displaystyle \displaystyle {={\frac {1}{4}}\pi \,\,{{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}+1{\Bigg )}}^{2}-{\frac {1}{2}}\pi \,\,{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}+1{\Bigg )}+1}}
=
1
4
π
c
a
r
d
Q
,
R
1
2
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
−
1
4
π
+
1
{\displaystyle \displaystyle {={\frac {1}{4}}\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}-{\frac {1}{4}}\pi +1}}
≠
c
a
r
d
Q
,
R
1
2
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
{\displaystyle \displaystyle {\neq {card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}
=
c
a
r
d
Q
,
R
2
(
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}
Fin du théorème
Début d’un théorème
NB : Matheux philosophe, c'est moi, Guillaume FOUCART.
[Citation de "Matheux philosophe"]
[Citation de "bolza"]
"L'infini" de l'intervalle
[
0
,
1
]
{\displaystyle [0,1]}
[
0
,
10
]
{\displaystyle [0,10]}
Là encore intuitivement je comprends parfaitement qu'on puisse penser "oui".
Et effectivement on pourrait se dire qu'il y a beaucoup plus de quantité de matière
dans un fil de
10
c
m
{\displaystyle 10\,\,cm}
1
c
m
{\displaystyle 1\,\,cm}
Le problème c'est que la quantité de matière dans un fil de
10
c
m
{\displaystyle 10\,\,cm}
1
c
m
{\displaystyle 1\,\,cm}
En effet, ils sont constitués d'un nombre fini d'atomes.
On compare donc ici deux ensembles finis dont un est plus grand que l'autre.
Mais entre ces atomes, il y a beaucoup de vide.
Pour que le fil corresponde exactement à la notion mathématiques d'intervalle,
il faudrait rajouter plein plein d'atomes pour combler ce vide et tous les relier entre eux,
et ce nombre d'atomes que l'on doit rajouter, c'est une infinité .
Et il se trouve que le nombre d'atomes à rajouter pour le fil de
10
c
m
{\displaystyle 10\,\,cm}
1
c
m
{\displaystyle 1\,\,cm}
(car, il y a une bijection entre
[
0
,
1
]
{\displaystyle [0,1]}
[
0
,
10
]
{\displaystyle [0,10]}
Une bijection ça veut dire que l'on a une correspondance un à un entre les éléments des deux ensembles)
Bon je ne sais pas si tout cela t'a convaincu, mais les intervalles
[
0
,
1
]
{\displaystyle [0,1]}
[
0
,
10
]
{\displaystyle [0,10]}
Ensuite tu peux très bien essayer de définir une fonction qui a des propriétés plus "intuitives"
sur la façon de "quantifier" les ensembles, mais je crois que cela existe déjà, ça s'appelle la "longueur".
En effet la longueur de l'intervalle
[
0
,
1
]
{\displaystyle [0,1]}
1
{\displaystyle 1}
[
0
,
10
]
{\displaystyle [0,10]}
10
{\displaystyle 10}
10
>
1
{\displaystyle 10>1}
En fait je crois que tu confonds les notions de "cardinalité" et de "grandeur".
P.S : Pour bien comprendre la différence, imagine un fil élastique.
Tu tends le fil de façon à ce qu'il ait une longueur de
1
c
m
{\displaystyle 1\,\,cm}
10
c
m
{\displaystyle 10\,\,cm}
1
{\displaystyle 1}
10
c
m
{\displaystyle 10\,\,cm}
[Fin Citation de "bolza"]
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
NB : Le cas d'une classe de parties bornées de
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle \mathbb {R} ^{n}}
C
0
{\displaystyle {\mathcal {C}}^{0}}
C
1
{\displaystyle {\mathcal {C}}^{1}}
∀
i
∈
N
n
∗
,
R
i
{\displaystyle \forall i\in \mathbb {N} _{n}^{*},\,\,{\cal {R}}_{i}}
R
i
{\displaystyle \mathbb {R} ^{i}}
O
i
(
0
)
j
∈
N
i
∗
{\displaystyle O_{i}{(0)}_{j\in \mathbb {N} _{i}^{*}}}
[
0
,
10
[
=
⨆
i
∈
N
10
∗
[
i
−
1
,
i
[
{\displaystyle \displaystyle {[0,10[=\bigsqcup _{i\in {\mathbb {N} }_{10}^{*}}[i-1,i[}}
c
a
r
d
Q
,
R
1
(
[
0
,
10
[
)
=
c
a
r
d
Q
,
R
1
(
⨆
i
∈
N
10
∗
[
i
−
1
,
i
[
)
=
∑
i
∈
N
10
∗
c
a
r
d
Q
,
R
1
(
[
i
−
1
,
i
[
)
=
∑
i
∈
N
10
∗
c
a
r
d
Q
,
R
1
(
[
0
,
1
[
)
=
c
a
r
d
Q
,
R
1
(
[
0
,
1
[
)
∑
i
∈
N
10
∗
1
=
10
c
a
r
d
Q
,
R
1
(
[
0
,
1
[
)
≠
c
a
r
d
Q
,
R
1
(
[
0
,
1
[
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{1}}([0,10[)={card}_{Q,{\cal {R}}_{1}}(\bigsqcup _{i\in {\mathbb {N} }_{10}^{*}}[i-1,i[)=\sum _{i\in {\mathbb {N} }_{10}^{*}}{card}_{Q,{\cal {R}}_{1}}([i-1,i[)=\sum _{i\in {\mathbb {N} }_{10}^{*}}{card}_{Q,{\cal {R}}_{1}}([0,1[)={card}_{Q,{\cal {R}}_{1}}([0,1[)\sum _{i\in {\mathbb {N} }_{10}^{*}}1=10\,\,{card}_{Q,{\cal {R}}_{1}}([0,1[)\neq {card}_{Q,{\cal {R}}_{1}}([0,1[)}}
c
a
r
d
P
(
[
0
,
10
[
)
=
c
a
r
d
P
(
⨆
i
∈
N
10
∗
[
i
−
1
,
i
[
)
=
∑
i
∈
N
10
∗
c
a
r
d
P
(
[
i
−
1
,
i
[
)
=
∑
i
∈
N
10
∗
c
a
r
d
P
(
[
0
,
1
[
)
=
c
a
r
d
P
(
[
0
,
1
[
)
∑
i
∈
N
10
∗
1
=
10
c
a
r
d
P
(
[
0
,
1
[
)
=
c
a
r
d
P
(
[
0
,
1
[
)
{\displaystyle \displaystyle {{card}_{P}([0,10[)={card}_{P}(\bigsqcup _{i\in {\mathbb {N} }_{10}^{*}}[i-1,i[)=\sum _{i\in {\mathbb {N} }_{10}^{*}}{card}_{P}([i-1,i[)=\sum _{i\in {\mathbb {N} }_{10}^{*}}{card}_{P}([0,1[)={card}_{P}([0,1[)\,\,\sum _{i\in {\mathbb {N} }_{10}^{*}}1=10\,\,{card}_{P}([0,1[)={card}_{P}([0,1[)}}
On considère le plafonnement carré de
R
2
{\displaystyle \mathbb {R} ^{2}}
O
2
{\displaystyle O_{2}}
R
2
{\displaystyle {\cal {R}}_{2}}
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
{\displaystyle {\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}}
Dans ce qui suit, où les intégrales sont encore à définir et
c
a
r
d
Q
{\displaystyle {card}_{Q}}
Cf. pour la définition de certains termes et le détail de certains calculs :
"2 calculs de la F-quantité de
R
2
{\displaystyle \mathbb {R} ^{2}}
|
{\displaystyle |}
R
2
{\displaystyle \mathbb {R} ^{2}}
O
2
(
0
,
0
)
{\displaystyle O_{2}(0,0)}
R
2
{\displaystyle {\cal {R}}_{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
c
a
r
d
Q
,
R
2
(
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
)
=
c
a
r
d
Q
,
R
2
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
2
)
=
(
c
a
r
d
Q
,
R
1
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
)
2
>
c
a
r
d
Q
,
R
1
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}={{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}{\Bigg )}}^{2}>{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
=
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
2
=
⨆
x
∈
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
{
(
x
,
y
)
∈
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
2
|
y
∈
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
}
{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}={{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}=\bigsqcup _{x\in {\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}{\bigg \{}(x,y)\in {{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}{\bigg |}y\in {\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg \}}}}
R
1
=
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
{\displaystyle \displaystyle {R_{1}={\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}}
R
1
2
=
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
2
{\displaystyle \displaystyle {R_{1}^{2}={{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}}}
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
=
R
1
2
=
⨆
x
∈
R
1
{
(
x
,
y
)
∈
R
1
2
|
y
∈
R
1
}
{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}=R_{1}^{2}=\bigsqcup _{x\in R_{1}}\{(x,y)\in R_{1}^{2}|y\in R_{1}\}}}
c
a
r
d
Q
,
R
2
(
[
R
2
,
(
[
−
r
,
r
]
2
)
r
∈
N
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}
=
c
a
r
d
Q
,
R
2
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
2
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}}}
=
c
a
r
d
Q
,
R
2
(
R
1
2
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}(R_{1}^{2})}}
=
c
a
r
d
Q
,
R
2
(
⨆
x
∈
R
1
{
(
x
,
y
)
∈
R
1
2
|
y
∈
R
1
}
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}{\Big (}\bigsqcup _{x\in R_{1}}\{(x,y)\in R_{1}^{2}|y\in R_{1}\}{\Big )}}}
=
∫
R
1
c
a
r
d
Q
,
R
2
(
{
(
x
,
y
)
∈
R
1
2
|
y
∈
R
1
}
)
d
c
a
r
d
Q
,
R
1
(
x
)
{\displaystyle \displaystyle {=\int _{R_{1}}{card}_{Q,{\cal {R}}_{2}}{\Big (}\{(x,y)\in R_{1}^{2}|y\in R_{1}\}{\Big )}\,\,d\,\,{card}_{Q,{\cal {R}}_{1}}(x)}}
=
∫
R
1
c
a
r
d
Q
,
R
1
(
R
1
)
d
c
a
r
d
Q
,
R
1
(
x
)
{\displaystyle \displaystyle {=\int _{R_{1}}{card}_{Q,{\cal {R}}_{1}}(R_{1})\,\,d\,\,{card}_{Q,{\cal {R}}_{1}}(x)}}
=
c
a
r
d
Q
,
R
1
(
R
1
)
∫
R
1
d
c
a
r
d
Q
,
R
1
(
x
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}(R_{1})\,\,\int _{R_{1}}\,\,d\,\,{card}_{Q,{\cal {R}}_{1}}(x)}}
=
c
a
r
d
Q
,
R
1
(
R
1
)
c
a
r
d
Q
,
R
1
(
R
1
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}(R_{1})\,\,{card}_{Q,{\cal {R}}_{1}}(R_{1})}}
=
(
c
a
r
d
Q
,
R
1
(
R
1
)
)
2
{\displaystyle \displaystyle {={{\Big (}{card}_{Q,{\cal {R}}_{1}}(R_{1}){\Big )}}^{2}}}
=
c
a
r
d
Q
,
R
1
2
(
R
1
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}^{2}(R_{1})}}
>
c
a
r
d
Q
,
R
1
(
R
1
)
{\displaystyle \displaystyle {>{card}_{Q,{\cal {R}}_{1}}(R_{1})}}
=
c
a
r
d
Q
,
R
1
(
[
R
,
(
[
−
r
,
r
]
)
r
∈
N
]
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}
c
a
r
d
P
(
R
2
)
=
(
c
a
r
d
P
(
R
)
)
2
=
c
a
r
d
P
(
R
)
{\displaystyle \displaystyle {{card}_{P}({\mathbb {R} }^{2})={{\Big (}{card}_{P}(\mathbb {R} ){\Big )}}^{2}={card}_{P}(\mathbb {R} )}}
R
{\displaystyle \mathbb {R} }
[
0
,
1
]
{\displaystyle [0,1]}
R
2
{\displaystyle {\mathbb {R} }^{2}}
[
0
,
1
]
2
{\displaystyle {[0,1]}^{2}}
ou plus simple :
c
a
r
d
Q
,
R
2
(
[
N
2
,
(
N
⋂
[
0
,
p
]
)
2
p
∈
N
]
)
=
c
a
r
d
Q
,
R
2
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
2
)
=
(
c
a
r
d
Q
,
R
1
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
)
)
2
>
c
a
r
d
Q
,
R
1
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\Bigg (}{\bigg [}\mathbb {N} ^{2},{{(\mathbb {N} \bigcap [0,p])}^{2}}_{p\in \mathbb {N} }{\bigg ]}{\Bigg )}={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}={{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}{\Bigg )}}^{2}>{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}}
[
N
2
,
(
N
⋂
[
0
,
p
]
)
2
p
∈
N
]
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
2
=
⨆
n
∈
N
{
(
n
,
m
)
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
2
|
m
∈
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
}
{\displaystyle \displaystyle {{\bigg [}\mathbb {N} ^{2},{{(\mathbb {N} \bigcap [0,p])}^{2}}_{p\in \mathbb {N} }{\bigg ]}={{\bigg [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\bigg ]}}^{2}=\bigsqcup _{n\in \mathbb {N} }{\Bigg \{}(n,m)\in {{\bigg [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\bigg ]}}^{2}{\Bigg |}m\in {\bigg [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\bigg ]}{\Bigg \}}}}
N
1
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
{\displaystyle \displaystyle {N_{1}={\bigg [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\bigg ]}}}
N
1
2
=
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
2
{\displaystyle \displaystyle {N_{1}^{2}={{\bigg [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\bigg ]}}^{2}}}
[
N
2
,
(
N
⋂
[
0
,
p
]
)
2
p
∈
N
]
=
N
1
2
=
⨆
n
∈
N
1
{
(
n
,
m
)
∈
N
1
2
|
m
∈
N
1
}
{\displaystyle \displaystyle {{\bigg [}\mathbb {N} ^{2},{{(\mathbb {N} \bigcap [0,p])}^{2}}_{p\in \mathbb {N} }{\bigg ]}=N_{1}^{2}=\bigsqcup _{n\in N_{1}}\{(n,m)\in N_{1}^{2}|m\in N_{1}\}}}
c
a
r
d
Q
,
R
2
(
[
N
2
,
(
N
⋂
[
0
,
p
]
)
2
p
∈
N
]
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\Bigg (}{{\bigg [}\mathbb {N} ^{2},{(\mathbb {N} \bigcap [0,p])}^{2}}_{p\in \mathbb {N} }{\bigg ]}{\Bigg )}}}
=
c
a
r
d
Q
,
R
2
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
2
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}}}
=
c
a
r
d
Q
,
R
2
(
N
1
2
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}(N_{1}^{2})}}
=
c
a
r
d
Q
,
R
2
(
⨆
n
∈
N
1
{
(
n
,
m
)
∈
N
1
2
|
m
∈
N
1
}
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}{\Big (}\bigsqcup _{n\in N_{1}}\{(n,m)\in N_{1}^{2}|m\in N_{1}\}{\Big )}}}
=
∑
n
∈
N
1
c
a
r
d
Q
,
R
2
(
{
(
n
,
m
)
∈
N
1
2
|
m
∈
N
1
}
)
{\displaystyle \displaystyle {=\sum _{n\in N_{1}}{card}_{Q,{\cal {R}}_{2}}{\Big (}\{(n,m)\in N_{1}^{2}|m\in N_{1}\}{\Big )}}}
=
∑
n
∈
N
1
c
a
r
d
Q
,
R
1
(
N
1
)
{\displaystyle \displaystyle {=\sum _{n\in N_{1}}{card}_{Q,{\cal {R}}_{1}}(N_{1})}}
=
c
a
r
d
Q
,
R
1
(
N
1
)
∑
n
∈
N
1
1
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}(N_{1})\,\,\sum _{n\in N_{1}}1}}
=
c
a
r
d
Q
,
R
1
(
N
1
)
c
a
r
d
Q
,
R
1
(
N
1
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}(N_{1})\,\,{card}_{Q,{\cal {R}}_{1}}(N_{1})}}
=
(
c
a
r
d
Q
,
R
1
(
N
1
)
)
2
{\displaystyle \displaystyle {={{\Big (}{card}_{Q,{\cal {R}}_{1}}(N_{1}){\Big )}}^{2}}}
=
c
a
r
d
Q
,
R
1
2
(
N
1
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}^{2}(N_{1})}}
>
c
a
r
d
Q
,
R
1
(
N
1
)
{\displaystyle \displaystyle {>{card}_{Q,{\cal {R}}_{1}}(N_{1})}}
=
c
a
r
d
Q
,
R
1
(
[
N
,
(
N
⋂
[
0
,
p
]
)
p
∈
N
]
)
{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}}
c
a
r
d
P
(
N
2
)
=
(
c
a
r
d
P
(
N
)
)
2
=
c
a
r
d
P
(
N
)
{\displaystyle \displaystyle {{card}_{P}({\mathbb {N} }^{2})={{\Big (}{card}_{P}(\mathbb {N} ){\Big )}}^{2}={card}_{P}(\mathbb {N} )}}
E
′
∈
P
V
(
R
n
)
{\displaystyle E'\in {PV}({\mathbb {R} }^{n})}
∀
x
∈
E
′
,
A
x
∈
P
V
(
R
n
)
{\displaystyle \forall x\in E',\,\,A_{x}\in {PV}({\mathbb {R} }^{n})}
∀
x
,
y
∈
E
′
,
x
≠
y
,
A
x
⋂
A
y
=
∅
{\displaystyle \displaystyle {\forall x,y\in E',\,\,x\neq y,\,\,A_{x}\bigcap A_{y}=\emptyset }}
A
=
⨆
x
∈
E
′
A
x
{\displaystyle \displaystyle {A=\bigsqcup _{x\in E'}A_{x}}}
alors
c
a
r
d
Q
,
R
n
(
A
)
=
c
a
r
d
Q
,
R
n
(
⨆
x
∈
E
′
A
x
)
=
∫
E
′
c
a
r
d
Q
,
R
n
(
A
x
)
d
c
a
r
d
Q
,
R
n
(
x
)
{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{n}}(A)={card}_{Q,{\cal {R}}_{n}}{\Big (}\bigsqcup _{x\in E'}A_{x}{\Big )}=\int _{E'}{card}_{Q,{\cal {R}}_{n}}(A_{x})\,\,d\,\,{card}_{Q,{\cal {R}}_{n}}(x)}}
(
∗
)
c
a
r
d
P
(
A
)
=
c
a
r
d
P
(
⨆
x
∈
E
′
A
x
)
=
∫
E
′
c
a
r
d
P
(
A
x
)
d
c
a
r
d
Q
,
R
n
(
x
)
{\displaystyle \displaystyle {(*)\,\,{card}_{P}(A)={card}_{P}{\Big (}\bigsqcup _{x\in E'}A_{x}{\Big )}=\int _{E'}{card}_{P}(A_{x})\,\,d\,\,{card}_{Q,{\cal {R}}_{n}}(x)}}
∃
E
″
∈
P
(
E
′
)
:
E
″
=
{
x
∈
E
′
,
A
x
≠
∅
}
{\displaystyle \displaystyle {\exists E''\in {\cal {P}}(E')\,\,:\,\,E''=\{x\in E',\,\,A_{x}\neq \emptyset \}}}
A
=
⨆
x
∈
E
″
A
x
{\displaystyle \displaystyle {A=\bigsqcup _{x\in E''}A_{x}}}
(24-06-2021 : Rectification : La notion de F-quantité n'est pas a priori une mesure définie sur
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
P
V
(
R
n
)
{\displaystyle {PV}(\mathbb {R} ^{n})}
Dans la suite de ce message, il y a vraisemblablement quelques précautions à prendre [et peut-être même dans ce qui précède concernant les égalités
(
∗
)
{\displaystyle (*)}
Une égalité n'impliquant que des F-quantités ou que des cardinaux potentiels, n'a pas le même sens et la même interprétation qu'une égalité impliquant à la fois le cardinal potentiel et la F-quantité.
c
a
r
d
P
(
R
2
)
=
c
a
r
d
P
(
R
)
{\displaystyle \displaystyle {{card}_{P}(\mathbb {R} ^{2})={card}_{P}(\mathbb {R} )}}
c
a
r
d
P
(
R
2
)
=
c
a
r
d
P
(
⨆
x
∈
R
{
(
x
,
y
)
∈
R
2
|
y
∈
R
}
)
=
∫
R
c
a
r
d
P
(
{
(
x
,
y
)
∈
R
2
|
y
∈
R
}
)
d
c
a
r
d
Q
,
R
1
(
x
)
=
∫
R
c
a
r
d
P
(
R
)
d
c
a
r
d
Q
,
R
1
(
x
)
{\displaystyle {card}_{P}({\mathbb {R} }^{2})={card}_{P}(\bigsqcup _{x\in \mathbb {R} }\{(x,y)\in {\mathbb {R} }^{2}|y\in \mathbb {R} \})=\int _{\mathbb {R} }{card}_{P}(\{(x,y)\in {\mathbb {R} }^{2}|y\in \mathbb {R} \})\,\,d\,\,{card}_{Q,{\cal {R}}_{1}}(x)=\int _{\mathbb {R} }{card}_{P}(\mathbb {R} )\,\,d\,\,{card}_{Q,{\cal {R}}_{1}}(x)}
=
c
a
r
d
P
(
R
)
∫
R
d
c
a
r
d
Q
,
R
1
(
x
)
=
c
a
r
d
P
(
R
)
c
a
r
d
Q
,
R
1
(
R
)
{\displaystyle \displaystyle {={card}_{P}(\mathbb {R} )\,\,\int _{\mathbb {R} }\,\,d\,\,{card}_{Q,{\cal {R}}_{1}}(x)={card}_{P}(\mathbb {R} )\,\,{card}_{Q,{\cal {R}}_{1}}(\mathbb {R} )}}
c
a
r
d
P
(
R
)
=
c
a
r
d
P
(
R
)
c
a
r
d
Q
,
R
1
(
R
)
{\displaystyle \displaystyle {{card}_{P}(\mathbb {R} )={card}_{P}(\mathbb {R} )\,\,{card}_{Q,{\cal {R}}_{1}}(\mathbb {R} )}}
[Fin de Citation de "Matheux philosophe"]
Fin du théorème
|
{\displaystyle |}
R
n
{\displaystyle \mathbb {R} ^{n}}
O
n
{\displaystyle O_{n}}
R
n
{\displaystyle {\cal {R_{n}}}}
R
n
{\displaystyle \mathbb {R} ^{n}}
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
[ modifier | modifier le wikicode ] Début d’un théorème
Soit
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
l'espace
R
n
{\displaystyle \mathbb {R} ^{n}}
R
n
{\displaystyle {\cal {R}}_{n}}
O
n
(
0
)
i
∈
N
n
∗
{\displaystyle O_{n}{(0)}_{i\in \mathbb {N} _{n}^{*}}}
[
R
n
,
(
B
R
n
(
O
n
,
r
)
¯
)
r
∈
N
]
=
lim
r
∈
N
,
r
→
sup
(
N
)
B
R
n
(
O
n
,
r
)
¯
{\displaystyle \displaystyle {{\bigg [}\mathbb {R} ^{n},{{\Big (}{\overline {B_{\mathbb {R} ^{n}}(O_{n},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\overline {B_{\mathbb {R} ^{n}}(O_{n},r)}}}}
∀
M
,
M
′
∈
R
n
,
M
≠
O
n
,
M
′
≠
O
n
,
c
a
r
d
Q
,
R
n
(
(
O
n
M
)
)
=
c
a
r
d
Q
,
R
n
(
(
O
n
M
′
)
)
=
c
a
r
d
Q
,
R
1
(
R
)
{\displaystyle \forall M,M'\in \mathbb {R} ^{n},\,\,M\neq O_{n},\,\,M'\neq O_{n},\,\,card_{Q,{\cal {R_{n}}}}{\Big (}(O_{n}M){\Big )}=card_{Q,{\cal {R_{n}}}}{\Big (}(O_{n}M'){\Big )}=card_{Q,{\cal {R_{1}}}}(\mathbb {R} )}
et
∀
M
,
M
′
∈
R
n
,
M
≠
O
n
,
M
′
≠
O
n
,
c
a
r
d
Q
,
R
n
(
[
O
n
M
)
)
=
c
a
r
d
Q
,
R
n
(
[
O
n
M
′
)
)
=
c
a
r
d
Q
,
R
1
(
R
+
)
=
1
2
c
a
r
d
Q
,
R
1
(
R
)
+
1
2
{\displaystyle \forall M,M'\in \mathbb {R} ^{n},\,\,M\neq O_{n},\,\,M'\neq O_{n},\,\,card_{Q,{\cal {R_{n}}}}{\Big (}[O_{n}M){\Big )}=card_{Q,{\cal {R_{n}}}}{\Big (}[O_{n}M'){\Big )}=card_{Q,{\cal {R_{1}}}}(\mathbb {R} _{+})={\frac {1}{2}}card_{Q,{\cal {R_{1}}}}(\mathbb {R} )+{\frac {1}{2}}}
=
1
2
c
a
r
d
Q
,
R
n
(
(
O
n
M
)
)
+
1
2
{\displaystyle ={\frac {1}{2}}card_{Q,{\cal {R_{n}}}}{\Big (}(O_{n}M){\Big )}+{\frac {1}{2}}}
∀
A
∈
R
n
,
A
≠
O
n
,
∀
M
∈
R
n
,
M
≠
O
n
,
M
≠
A
,
c
a
r
d
Q
,
R
n
(
(
A
M
)
)
≠
c
a
r
d
Q
,
R
n
(
(
O
n
M
)
)
{\displaystyle \forall A\in \mathbb {R} ^{n},\,\,A\neq O_{n},\,\,\forall M\in \mathbb {R} ^{n},\,\,M\neq O_{n},\,\,M\neq A,\,\,card_{Q,{\cal {R_{n}}}}{\Big (}(AM){\Big )}\neq card_{Q,{\cal {R_{n}}}}{\Big (}(O_{n}M){\Big )}}
et même
c
a
r
d
Q
,
R
n
(
[
A
M
)
)
<
c
a
r
d
Q
,
R
n
(
(
A
M
)
)
<
c
a
r
d
Q
,
R
n
(
(
O
n
M
)
)
{\displaystyle {card}_{Q,{\cal {R}}_{n}}{\Big (}[AM){\Big )}<{card}_{Q,{\cal {R}}_{n}}{\Big (}(AM){\Big )}<{card}_{Q,{\cal {R}}_{n}}{\Big (}(O_{n}M){\Big )}}
et
∀
A
,
B
∈
R
n
,
A
≠
O
n
,
B
≠
O
n
,
A
≠
B
,
∀
M
∈
R
n
,
M
≠
O
n
,
c
a
r
d
Q
,
R
n
(
(
A
B
)
)
≠
c
a
r
d
Q
,
R
n
(
(
O
n
M
)
)
{\displaystyle \forall A,B\in \mathbb {R} ^{n},\,\,A\neq O_{n},\,\,B\neq O_{n},\,\,A\neq B,\,\,\forall M\in \mathbb {R} ^{n},\,\,M\neq O_{n},\,\,card_{Q,{\cal {R_{n}}}}{\Big (}(AB){\Big )}\neq card_{Q,{\cal {R_{n}}}}{\Big (}(O_{n}M){\Big )}}
et même
c
a
r
d
Q
,
R
n
(
[
A
B
)
)
<
c
a
r
d
Q
,
R
n
(
(
A
B
)
)
<
c
a
r
d
Q
,
R
n
(
(
O
n
M
)
)
{\displaystyle card_{Q,{\cal {R_{n}}}}{\Big (}[AB){\Big )}<card_{Q,{\cal {R_{n}}}}{\Big (}(AB){\Big )}<card_{Q,{\cal {R_{n}}}}{\Big (}(O_{n}M){\Big )}}
∀
A
∈
R
n
,
A
≠
O
n
,
∀
M
∈
R
n
,
M
≠
O
n
,
M
≠
A
,
{\displaystyle \forall A\in \mathbb {R} ^{n},\,\,A\neq O_{n},\,\,\forall M\in \mathbb {R} ^{n},\,\,M\neq O_{n},\,\,M\neq A,}
c
a
r
d
Q
,
R
n
(
[
A
M
)
)
<
c
a
r
d
Q
,
R
n
(
[
O
n
M
)
)
{\displaystyle card_{Q,{\cal {R_{n}}}}{\Big (}[AM){\Big )}<card_{Q,{\cal {R_{n}}}}{\Big (}[O_{n}M){\Big )}}
c
a
r
d
Q
,
R
n
(
[
A
M
)
)
>
c
a
r
d
Q
,
R
n
(
[
O
n
M
)
)
{\displaystyle card_{Q,{\cal {R_{n}}}}{\Big (}[AM){\Big )}>card_{Q,{\cal {R_{n}}}}{\Big (}[O_{n}M){\Big )}}
c
a
r
d
Q
,
R
n
(
[
A
M
)
)
=
c
a
r
d
Q
,
R
n
(
[
O
n
M
)
)
{\displaystyle card_{Q,{\cal {R_{n}}}}{\Big (}[AM){\Big )}=card_{Q,{\cal {R_{n}}}}{\Big (}[O_{n}M){\Big )}}
∀
A
,
B
∈
R
n
,
A
≠
O
n
,
B
≠
O
n
,
A
≠
B
,
∀
M
∈
R
n
,
M
≠
O
n
,
{\displaystyle \forall A,B\in \mathbb {R} ^{n},\,\,A\neq O_{n},\,\,B\neq O_{n},\,\,A\neq B,\,\,\forall M\in \mathbb {R} ^{n},\,\,M\neq O_{n},}
c
a
r
d
Q
,
R
n
(
[
A
B
)
)
<
c
a
r
d
Q
,
R
n
(
[
O
n
M
)
)
{\displaystyle {card}_{Q,{\cal {R}}_{n}}{\Big (}[AB){\Big )}<{card}_{Q,{\cal {R}}_{n}}{\Big (}[O_{n}M){\Big )}}
c
a
r
d
Q
,
R
n
(
[
A
B
)
)
>
c
a
r
d
Q
,
R
n
(
[
O
n
M
)
)
{\displaystyle {card}_{Q,{\cal {R}}_{n}}{\Big (}[AB){\Big )}>{card}_{Q,{\cal {R}}_{n}}{\Big (}[O_{n}M){\Big )}}
c
a
r
d
Q
,
R
n
(
[
A
B
)
)
=
c
a
r
d
Q
,
R
n
(
[
O
n
M
)
)
{\displaystyle {card}_{Q,{\cal {R}}_{n}}{\Big (}[AB){\Big )}={card}_{Q,{\cal {R}}_{n}}{\Big (}[O_{n}M){\Big )}}
Fin du théorème
Remarque importante :
B
{\displaystyle B}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
au lieu de parler de la F-quantité relative au repère
R
{\displaystyle {\mathcal {R}}}
B
{\displaystyle B}
c
a
r
d
Q
,
R
(
B
)
{\displaystyle {card}_{Q,{\mathcal {R}}}(B)}
R
{\displaystyle {\mathcal {R}}}
[
A
,
(
A
i
)
i
∈
I
]
{\displaystyle [A,{(A_{i})}_{i\in I}]}
B
{\displaystyle B}
c
a
r
d
Q
,
R
,
[
A
,
(
A
i
)
i
∈
I
]
(
B
)
{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(B)}
et dans ce cas on a : "
c
a
r
d
Q
,
R
,
[
A
,
(
A
i
)
i
∈
I
]
(
B
)
=
d
e
´
f
c
a
r
d
Q
,
R
(
B
⋂
[
A
,
(
A
i
)
i
∈
I
]
)
=
c
a
r
d
Q
,
R
(
[
B
⋂
A
,
(
B
⋂
A
i
)
i
∈
I
]
)
{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(B){\underset {d{\acute {e}}f}{=}}{card}_{Q,{\mathcal {R}}}{\Big (}B\bigcap [A,{(A_{i})}_{i\in I}]{\Big )}={card}_{Q,{\mathcal {R}}}{\Big (}{\Big [}B\bigcap A,{{\Big (}B\bigcap A_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}}
et, en particulier, on a : "
c
a
r
d
Q
,
R
,
[
A
,
(
A
i
)
i
∈
I
]
(
A
)
=
d
e
´
f
![{\displaystyle [0;1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3bf59a5da5d8181083b228c8933efbda133483)
![{\displaystyle [-1,1]\subsetneq [-2,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10d13a0d7daaa05253e19368418be7be2486383d)
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle [-2,2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f94b820404eca2a458cb2c7d8c24be85fffccf90)
![{\displaystyle \displaystyle {{\frac {{card}_{Q}([-2,2]\setminus \{0\})}{{card}_{Q}([-1,1]\setminus \{0\})}}={\frac {{card}_{Q}([-2,2])-{card}_{Q}(\{0\})}{{card}_{Q}([-1,1])-{card}_{Q}(\{0\})}}={\frac {{card}_{Q}([-2,2])-1}{{card}_{Q}([-1,1])-1}}=2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f622bd8ad9743b76549a18c43279d6eba5d460d)
![{\displaystyle {card}_{Q}([-1,1])<{card}_{Q}([-2,2])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6596575904d4d3408812888a97a40996de6da4)
![{\displaystyle {card}_{P}([-2,2])={card}([-2,2])={card}([-1,1])={card}_{P}([-1,1])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22ad45fb6b20cbb0e0e56fd76cc47572517f843e)
![{\displaystyle i\in [\![1,n]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c593442211b1a25dc31ca4d15ab8121fcb6b1034)
![{\displaystyle i\in [\![0,n]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b95feed3edf6db2ad337ff7f80201428104d7dfe)
![{\displaystyle j\in [\![0,i]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07a78e8abe1e8d972eb2d4cd36640a0ca23a7d26)
![{\displaystyle \displaystyle {{F_{0}}'\in {\mathcal {P}}{\bigg (}\mathbb {R} ^{n}\setminus {\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5caa94247d15b87e110b1ab334b54c862d9958)
![{\displaystyle j\in [\![1,i]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98a420840479184d6f9baab8200d81f84989e5f3)
![{\displaystyle \displaystyle {{F_{0,0}}'\in {\mathcal {P}}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16990e1a49d75ea16d9a5e52e89725ba7028d7b4)
![{\displaystyle \forall i\in [\![0,n]\!],\exists F_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c92d478f397f59bbf50ff942cf6fa3b085872799)
![{\displaystyle \forall i\in [\![0,n]\!],\,\,\forall j\in [\![0,i]\!],\,\,\exists F_{i,j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15dbf463b4d6d49d202f65fb32b7b4839c429428)
![{\displaystyle \displaystyle {A={\Big (}\bigsqcup _{i\in [\![0,n]\!]}F_{i}{\Big )}\setminus {\Big (}\bigsqcup _{i\in [\![0,n]\!]}\bigsqcup _{j\in [\![0,i]\!]}F_{i,j}{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68c2e8f95c44834374bcf1e548ff72e005323a05)
![{\displaystyle \forall i\in [\![1,n]\!],\exists U_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74652134d9a48e95f64ef4ecc88fb4acbb0a232e)
![{\displaystyle \forall i\in [\![1,n]\!],\,\,\forall j\in [\![1,i]\!],\,\,\exists U_{i,j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eef9ad16e07fad17ecb01fc6a3b79ca5fb483844)
![{\displaystyle \displaystyle {\exists {F_{0}}'\in {\mathcal {P}}{\bigg (}\mathbb {R} ^{n}\setminus {\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97181e613e65fb2d3db58895559bd4ed41a56dfd)
![{\displaystyle \displaystyle {\exists {F_{0,0}}'\in {\mathcal {P}}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eef566301b3203f4cd0b7543e37ce800cca2f9c)
![{\displaystyle \displaystyle {A={\bigg (}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}U_{i}{\Big )}\bigsqcup {F_{0}}'{\bigg )}\setminus {\bigg (}{\Big (}\bigsqcup _{i\in [\![1,n]\!]}\bigsqcup _{j\in [\![1,i]\!]}U_{i,j}{\Big )}\bigsqcup {F_{0,0}}'{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c9c18be8641616d889ce53239c187e337eb717)
![{\displaystyle i\,\,(i\in [\![0,n]\!])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47869cdb9e66a8d4c4ecd3704afafd5035a2912d)
![{\displaystyle <card_{Q,{\cal {R}}}(\{O\}\times ]-1,1[)<{card}_{Q,{\cal {R}}}(\{O\}\times [-1,1])<{card}_{Q,{\cal {R}}}(\{O\}\times [-2,2])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b965ca3f7f9a5b19a8c0ba79af8f3cbfa6a4922b)
![{\displaystyle ={card}_{Q,{\cal {R}}}{\Big (}\{O\}\times ([-2,2]+1){\Big )}<{card}_{Q,{\cal {R}}}{\bigg (}\{O\}\times {\Big (}([-2,2]+1)\bigsqcup \{4\}{\Big )}{\bigg )}<{card}_{Q,{\cal {R}}}{\Big (}\{O\}\times (\mathbb {R} \setminus [-2,2]){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5596dc944b2927cc50cd92ddec1314bb3258413)
![{\displaystyle <{card}_{Q,{\cal {R}}}{\Big (}\{O\}\times (\mathbb {R} \setminus [-1,1]){\Big )}<{card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {R} ^{*})<{card}_{Q,{\cal {R}}}(\{O\}\times \mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75d11d2ff6ffd6935a4d315423757921bbdf6bae)
![{\displaystyle <{card}_{Q,{\cal {R}}}([-1,1]\times [-1,1])<{card}_{Q,{\cal {R}}}([-2,2]\times [-2,2])<{card}_{Q,{\cal {R}}}(\mathbb {R} ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85f04eff0446acb881d84926c86f3f64f36f3ca8)
![{\displaystyle <{card}_{P}(\{O\}\times ]-1,1[)={card}_{P}(\{O\}\times [-1,1])={card}_{P}(\{O\}\times [-2,2])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3587eafd5bc00ab2db04861f649509bf312bfaf2)
![{\displaystyle ={card}_{P}{\Big (}\{O\}\times ([-2,2]+1){\Big )}={card}_{P}{\bigg (}\{O\}\times {\Big (}([-2,2]+1)\bigsqcup \{4\}{\Big )}{\bigg )}={card}_{P}{\Big (}\{O\}\times (\mathbb {R} \setminus [-2,2]){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5568497e847bd56c97efe9694071054135d4bb99)
![{\displaystyle ={card}_{P}{\Big (}\{O\}\times (\mathbb {R} \setminus [-1,1]){\Big )}={card}_{P}(\{O\}\times \mathbb {R} ^{*})={card}_{P}(\{O\}\times \mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d67a8b555448704bc2459f5613965b6a70097528)
![{\displaystyle ={card}_{P}([-1,1]\times [-1,1])={card}_{P}([-2,2]\times [-2,2])={card}_{P}(\mathbb {R} ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8888105718691a526ad52b1e97c60854aa49062e)
![{\displaystyle \displaystyle {{\underset {d{\acute {e}}f}{=}}{\Big \{}{P^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{P^{N}}\,\,polytope\,\,de\,\,\mathbb {R} ^{N}\,\,(c{\mbox{-}}{\grave {a}}{\mbox{-}}d\,\,poly{\grave {e}}dre\,\,compact,\,\,convexe\,\,[connexe]\,\,de\,\,\mathbb {R} ^{N}){\Big \}}\bigcup \{\emptyset \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca90c16a8c22b2df814efb8e199d576cd377e7a2)
![{\displaystyle F=\mathbb {R} _{n}[{card}_{Q,{\mathcal {R}}}(I)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57093ac0955a8d8797b4080dff3779dc52603c6e)
![{\displaystyle \displaystyle {\forall {(x_{m})}_{m\in \mathbb {N} }\subset {\mathbb {R} },\,\,convergente\,\,dans\,\,\mathbb {R} ,\,\,\lim _{m\rightarrow +\infty }{card}_{Q,{\cal {R}}}([0,x_{m}])={card}_{Q,{\cal {R}}}([0,\lim _{m\rightarrow +\infty }x_{m}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab69346278a13558783b7cc90ca7444376291e13)
![{\displaystyle ''[A,{(A_{i})}_{i\in I}]''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/788fac4b292ef5cc65dd0e8c5e9248623ed14877)
![{\displaystyle [A,{(A_{i})}_{i\in I}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b918504477f81083168b3858820cf9486b0cf5c)
![{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59df03935cabea6064b891db7dffecaedde222cb)
![{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(B){\underset {d{\acute {e}}f}{=}}{card}_{Q,{\mathcal {R}}}{\Big (}B\bigcap [A,{(A_{i})}_{i\in I}]{\Big )}={card}_{Q,{\mathcal {R}}}{\Big (}{\Big [}B\bigcap A,{{\Big (}B\bigcap A_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bae36f1baf3ec513e3132c332fc4b6a5872d80b)
![{\displaystyle {card}_{Q,{\mathcal {R}},[A,{(A_{i})}_{i\in I}]}(A){\underset {d{\acute {e}}f}{=}}{card}_{Q,{\mathcal {R}}}{\Big (}A\bigcap [A,{(A_{i})}_{i\in I}]{\Big )}={card}_{Q,{\mathcal {R}}}{\Big (}{\Big [}A,{{\Big (}A\bigcap A_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}{\underset {prop?}{=}}{card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7953de6eaf0abf47228617bbfde614bd2294fa)
![{\displaystyle {card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/876bcee9570669ca3e28ed0d52e64f6429f0bbb6)
![{\displaystyle [\mathbb {R} ^{n},{(A_{i})}_{i\in I}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cda8b2dfd2d79b4520fc37221397d02d16999f44)
![{\displaystyle \displaystyle {K_{0,{\stackrel {\circ }{I}}}=]0,{vol}^{1}({\stackrel {\circ }{I}})[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5192e0685c57d5a473a1c6b9d3dc4d35503a1c)
![{\displaystyle \displaystyle {K_{0,{\stackrel {\circ }{J}}}=]0,{vol}^{1}({\stackrel {\circ }{J}})[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e83e39ced8e90e573397ed82d06617f623b259bf)
![{\displaystyle \displaystyle {K_{0,{\overline {I}}}=[0,{vol}^{1}({\overline {I}})]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23e1e6372bd5b11739cdba90afe0562a4ed682e0)
![{\displaystyle \displaystyle {K_{0,{\overline {J}}}=[0,{vol}^{1}({\overline {J}})]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89194682c3cb2cd374930dadb4a9ade5e29cd0c7)
p,ip[)={card}_{Q,{\cal {R}}}(]0,p[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ecadd75079fbd9cd0908cb7b44e6b5113b5029)
p,ip[)={vol}^{1}(]0,p[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de711a7e69a39c8667821a080eec7db2b136228a)
p,ip[)+1={card}_{Q,{\cal {R}}}(]0,p[)+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8729414102f913040b4567f0ee4f007e7b0bd07e)
p,ip[)+{card}_{Q,{\cal {R}}}(\{(i-1)p\})={card}_{Q,{\cal {R}}}(]0,p[)+{card}_{Q,{\cal {R}}}(\{0\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/887f993d76d97be337338173077fa7f564b5905f)
p,ip[\bigsqcup \{(i-1)p\})={card}_{Q,{\cal {R}}}(]0,p[\bigsqcup \{0\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/657aebd4b98c5445b629bbfc8566d97c94a3da57)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(]0,kp[)={card}_{Q,{\cal {R}}}([0,kp[\setminus \{0\})={card}_{Q,{\cal {R}}}([0,kp[)-{card}_{Q,{\cal {R}}}(\{0\})=k\,\,{card}_{Q,{\cal {R}}}([0,p[)-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ead37bdf70a699d4e498899f4e180610eceb2e)
![{\displaystyle \displaystyle {=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[\bigsqcup \{0\}){\Big )}-1=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[)+{card}_{Q,{\cal {R}}}(\{0\}){\Big )}-1=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[)+1{\Big )}-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd759ac290b547346933e526fda67106a8da2f5)
![{\displaystyle \displaystyle {]0,kp[=(\bigsqcup _{i\in \mathbb {N} _{k}^{*}}](i-1)p,ip[)\bigsqcup (\bigsqcup _{i\in \mathbb {N} _{k-1}^{*}}\{ip\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69dd2ec34739aef32e498d9a51543c5864623d46)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(]0,kp[)=\sum _{i\in \mathbb {N} _{k}^{*}}{card}_{Q,{\mathcal {R}}}(](i-1)p,ip[)+\sum _{i\in \mathbb {N} _{k-1}^{*}}{card}_{Q,{\mathcal {R}}}(\{ip\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea032345a07d2e8bbb6bdcd3e24fb94ac8e6658)
p,ip[)={card}_{Q,{\mathcal {R}}}(]0,p[)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d68435213c4089114c07b22c3569def0598c6ee2)
p,ip[)=vol^{1}(]0,p[)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35cf2861abad3f6d923596520f6852d6e3780ec2)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(]0,kp[)=\sum _{i\in \mathbb {N} _{k}^{*}}{card}_{Q,{\mathcal {R}}}(]0,p[)+\sum _{i\in \mathbb {N} _{k-1}^{*}}{card}_{Q,{\mathcal {R}}}(\{p\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f33fbd027889d8be9e94ede3ec5af8ce385f1c1a)
![{\displaystyle {card}_{Q,{\mathcal {R}}}(]0,kp[)=k\,\,{card}_{Q,{\mathcal {R}}}(]0,p[)+(k-1)\,\,{card}_{Q,{\mathcal {R}}}(\{p\})=k\,\,{card}_{Q,{\mathcal {R}}}(]0,p[)+(k-1)=k\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,p[)+1{\Big )}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe65ab5762d75b180905ba44672ae74c4533b86)
![{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(]0,kp[)+1}{{card}_{Q,{\cal {R}}}(]0,p[)+1}}=k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06141daae07c6fdb49436f5d193cda22cc50c158)
![{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(]0,kp[)+1}{{card}_{Q,{\cal {R}}}(]0,p[)+1}}={\frac {{vol}^{1}(]0,kp[)}{{vol}^{1}(]0,p[)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8333bb89bbd2bac66d1537d1304f23e32fd6582a)
![{\displaystyle \displaystyle {{\underset {d{\acute {e}}f}{=}}{\Big \{}{P_{i}^{N}}\in {\cal {P}}(\mathbb {R} ^{N})\,\,{\Big |}\,\,{P_{i}^{N}}\,\,polytope\,\,de\,\,\mathbb {R} ^{N}\,\,(c{\mbox{-}}{\grave {a}}{\mbox{-}}d\,\,poly{\grave {e}}dre\,\,compact,\,\,convexe\,\,[connexe]\,\,de\,\,\mathbb {R} ^{N})\,\,et\,\,{dim}({P_{i}^{N}})=i{\Big \}}\bigcup \{\emptyset \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c2c30a7226182d640fa13764bf9c8767f09b98)
![{\displaystyle \displaystyle {={card}_{Q,2}{\Bigg (}\bigsqcup _{x\in [-1,1]}{\bigg [}{\Big (}x,-{\sqrt {1-x^{2}}}{\Big )},{\Big (}x,{\sqrt {1-x^{2}}}{\Big )}{\bigg ]}{\Bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a048e7a0c6503873ee795fafad51811354e5321)
![{\displaystyle \displaystyle {=\int _{[-1,1]}{card}_{Q,1}{\Bigg (}{\bigg [}{\Big (}x,-{\sqrt {1-x^{2}}}{\Big )},{\Big (}x,{\sqrt {1-x^{2}}}{\Big )}{\bigg ]}{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a238166027f83e64754ebe54d82eecf14ca7a04)
![{\displaystyle \displaystyle {=\int _{[-1,1]}{card}_{Q,1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big ]}{\bigg )}\,\,d\,\,{card}_{Q,1}(x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16d642bd4c9d2b71c39a09f7909c77a779098cf8)
![{\displaystyle \displaystyle {=\int _{]-1,1[}{card}_{Q,1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big ]}{\bigg )}\,\,d\,\,{card}_{Q,1}(x)+2\,\,{card}_{Q,1}(\{0\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9069c10071690b3a55ad52c378a6e2f042d71a)
![{\displaystyle \displaystyle {=\int _{]-1,1[}{\Bigg (}{card}_{Q,1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big [}{\bigg )}+1{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c697cda38463dbc2935af481b7fad4409b40b0c6)
![{\displaystyle \displaystyle {=\int _{]-1,1[}{\Bigg (}{vol}^{1}{\bigg (}{\Big [}-{\sqrt {1-x^{2}}},{\sqrt {1-x^{2}}}{\Big [}{\bigg )}\,\,{card}_{Q,1}([0,1[)+1{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b557ca8095ecbf6c32532b464d18a0e6708e5939)
![{\displaystyle \displaystyle {=\int _{]-1,1[}{\Bigg (}2{\sqrt {1-x^{2}}}\,\,{card}_{Q,1}([0,1[)+1{\Bigg )}\,\,d\,\,{card}_{Q,1}(x)+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acd26e73655640257291c15a8e86ecbc0d29da0c)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+\int _{]-1,1[}d\,\,{card}_{Q,1}(x)+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1725234d7fd8a3e366804bee48651c0173a71c9)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+{card}_{Q,1}(]-1,1[)+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c224f7b306cf6510ca5e78e4514ad98d31267b2a)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+{card}_{Q,1}([-1,1[)-1+2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1261fb4c497199aea27dd833ea441a637a841e)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+{vol}^{1}([-1,1[)\,\,{card}_{Q,1}([0,1[)+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e08686359477f6355923cb9cfd812d61dd5d6706)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+2\,\,{card}_{Q,1}([0,1[)+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4938187cbe8087203bbb3f5f3633fe3e1497c232)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,1}([0,1[)\,\,{\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+1{\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1e03441d84e9e7bf6d0109b7864c83e827036a)
![{\displaystyle \displaystyle {2\,\,{card}_{Q,1}([0,1[)\,\,{\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+1{\Big )}+1=\pi \,\,{card}_{Q,1}([0,1[)\,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f84c0185822a0f8de27963a394ce91cad8e23de)
![{\displaystyle \displaystyle {2\,\,{\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)+1{\Big )}=\pi \,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd3d48a579d0d94e856cfede8aeae55a24b5e68)
![{\displaystyle \displaystyle {\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)={\frac {\pi }{2}}\,\,{\Big (}{card}_{Q,1}([0,1[)+1{\Big )}-1={\frac {\pi }{2}}\,\,{card}_{Q,1}([0,1])-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/866fcc0425fccf86eb7f73a4222cc9da80527b62)
![{\displaystyle \displaystyle {\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)={\Big (}\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{vol}^{1}(x){\Big )}\,\,{card}_{Q,1}([0,1])-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9b93cb2f624491bc3ee7856265da607380b821)
![{\displaystyle \displaystyle {\int _{]-1,1[}{\sqrt {1-x^{2}}}\,\,d\,\,{card}_{Q,1}(x)={\frac {\pi }{2}}\,\,{card}_{Q,1}([0,1[)+{\frac {\pi }{2}}-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95cca1da385771a9e0a46621b76ff14bc2a5ed55)
![{\displaystyle ]-1,1[\not \in {PV}_{1}(\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34f3c49ef82b19cb0800010f66ef7ae85db2c640)
![{\displaystyle \displaystyle {{\frac {{card}_{Q}{\Big (}f({\overline {A}}){\Big )}-1}{{card}_{Q,1}([0,1])-1}}={\frac {{vol}^{1}{\Big (}f({\overline {A}}){\Big )}}{{vol}^{1}([0,1])}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/465880792bac09fc9029729dc7029536cab7493c)
![{\displaystyle \displaystyle {={\frac {{vol}^{1}{\Big (}f({\overline {A}}){\Big )}}{{vol}^{1}([0,1])}}\,\,{card}_{Q,1}([0,1])-{\frac {{vol}^{1}{\Big (}f({\overline {A}}){\Big )}}{{vol}^{1}([0,1])}}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c15bfedb19142b422417a837b2b9ac4ea3e1f4)
![{\displaystyle \displaystyle {={vol}^{1}{\Big (}f({\overline {A}}){\Big )}\,\,{card}_{Q,1}([0,1])-{vol}^{1}{\Big (}f({\overline {A}}){\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a0cbe3670a54ec866935c5794335e701e31232)
![{\displaystyle \displaystyle {A_{N}=\left[\bigsqcup _{i\in \mathbb {N} _{N}^{*}}{\Big (}A_{N-(i-1)}\setminus A_{N-i}{\Big )}\right]\bigsqcup A_{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4c363fe7f919f24373f3f0b7c68dbeef5e6228)
![{\displaystyle [A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}(\mathbb {R} ^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f248227d003a995cf613c8ad50289809427ca460)
![{\displaystyle {\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\in {\mathcal {P}}(\mathbb {R} ^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbaf47868aaf6d45e0396a47871c659bb5e539f)
![{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/330424159338c94dfcde4058c90c9eba26d67af4)
![{\displaystyle \displaystyle {\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,{\Big (}\bigcup _{i\in I}A_{i}=A\,\,si\,\,{(A_{i})}_{i\in I}\,\,\nearrow \,\,pour\,\,\subset ,\,\,ou,\,\,\bigcap _{i\in I}A_{i}=A\,\,si\,\,{(A_{i})}_{i\in I}\,\,\searrow \,\,pour\,\,\subset {\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc8e7b412ea97c358525d73b43fff4ecd0eb3ff)
![{\displaystyle [A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}({\mathbb {R} }^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}({\mathbb {R} }^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adabe42ea2bf13ff6ba8b01f8c65609767104edc)
![{\displaystyle \displaystyle {{\underset {i\in I,\,\,i\rightarrow \sup(I)}{{\text{lim}}_{classique}}}A_{i}=A\,\,\Leftrightarrow \,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/573cb64ba47eaf2bae7c39aee6a31c8e343b3329)
![{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}(I,{\mathcal {A}},{\mathcal {B}}){\underset {d{\acute {e}}f}{=}}{\Big \{}[A,{(A_{i})}_{i\in I}]\in {\mathcal {A}}\times {\mathcal {F}}(I,{\mathcal {B}})\,\,{\Big |}\,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}=[A,{(A_{i})}_{i\in I}]{\Big \}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1258c2ab38ed522a3beb5332fe28ec4a500e4e)
![{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}(I,{\mathcal {A}}){\underset {d{\acute {e}}f}{=}}{{\mathcal {P}}lafonnements}(I,{\mathcal {A}},{\mathcal {A}}){\underset {d{\acute {e}}f}{=}}{\Big \{}[A,{(A_{i})}_{i\in I}]\in {\mathcal {A}}\times {\mathcal {F}}(I,{\mathcal {A}})\,\,{\Big |}\,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}=[A,{(A_{i})}_{i\in I}]{\Big \}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abc7e72db284d5389b41aabb4fa1f16394f93b68)
![{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}={{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}),{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}={\Big \{}[A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}(\mathbb {R} ^{n})\times {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}\,\,{\Big |}\,\,\lim _{i\in I,\,\,i\rightarrow \sup(I)}A_{i}=[A,{(A_{i})}_{i\in I}]{\Big \}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/271b939745acbfab851e8a7c87c399ae34a38a12)
![{\displaystyle \displaystyle {\lim _{i\in I,i\rightarrow \sup(I)}A_{i}={\Big [}A,{(A_{i})}_{i\in I}{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6c9fa42752f892580c0ec47671ffc6c9c0a8e4)
![{\displaystyle card_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}=card_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}card_{Q,{\cal {R}}}(A_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f1934f02568eef74de1b260e5100e61425ba25)
![{\displaystyle {\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13290e499cdbb393ea840d66d2436f1a9d265d06)
![{\displaystyle \displaystyle {\lim _{i\in I,i\rightarrow \sup(I)}B_{i}={\Big [}A,{(B_{i})}_{i\in I}{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4df603b5cc942bcb909ff2babb4c83154815a02)
![{\displaystyle {\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\neq {\Big [}A,{(B_{i})}_{i\in I}{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b621c33c2ac60ec17becc7141b7effa0cd0d051)
![{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})\neq \lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}B_{i})={card}_{Q,{\cal {R}}}{\Big (}[A,(B_{i})_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b83b2b81f821c76386fb06688fc2a8087c8db27c)
![{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(A_{i})=\lim _{i\in I,\,\,i\to \sup(I)}{card}_{Q,{\cal {R}}}(B_{i})={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}B_{i})={card}_{Q,{\cal {R}}}{\Big (}[A,(B_{i})_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e518dec35bd3055b151a66456ae7b37fe468e867)
![{\displaystyle {card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}={card}_{Q,{\cal {R}}}(\lim _{i\in I,\,\,i\to \sup(I)}A_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9297fc9c47f1f8eb6fcbe47d66b22f84086feb70)
![{\displaystyle [A,(A_{i})_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}({\mathbb {R} }^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efe458fd5ac1c35a81f276e52867f0d0d1653a94)
![{\displaystyle [A,(A_{i})_{i\in I}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1679b3a8496e261e15e4b07bd7565b8f0602695)
![{\displaystyle {card}_{Q,{\cal {R}}}(A){\underset {d{\acute {e}}f}{=}}{card}_{Q,{\cal {R}}}{\Big (}[A,(A_{i})_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43f3341fcab918e78b0bc64f9f189a432d8d70e8)
![{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36684528553ef70cb40794f1cd85d2a8aca54156)
![{\displaystyle card_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}])\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd74fdd778f85e329ea5615fdf274da678f35f4d)
![{\displaystyle \forall [\emptyset ,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,\{\emptyset \},{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/360a484973f1a1f87ac61bb5935202848f1f6954)
![{\displaystyle {card}_{Q,{\cal {R}}}([\emptyset ,{(A_{i})}_{i\in I}])=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44e014f9d68d806341334a57067e1695e7b60bdb)
![{\displaystyle \forall x\in \mathbb {R} ^{n},\,\,\forall [\{x\},{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\Big \{}\{x\}{\Big \}},{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eacb892cb7a530a80178a41d2520282b5c309444)
![{\displaystyle {card}_{Q,{\cal {R}}}([\{x\},{(A_{i})}_{i\in I}])=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c889afbf7b1a03ccee6eb9a3124e0966f3673358)
![{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/470d5366dc90a8fcfe09ca0f0945913889d2cacb)
![{\displaystyle \displaystyle {{\Big [}A\bigcup B,{{\Big (}A_{i}\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]},{\Big [}A\bigcap B,{{\Big (}A_{i}\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd886ff0de69aa50cd80b63ab4a87b90a2041e11)
![{\displaystyle \forall [A,{(A_{i})}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f603ab2c76db2532e9b71a19d2816b322b407079)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A\bigcup B,{{\Big (}A_{i}\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}={card}_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}])+{card}_{Q,{\cal {R}}}([B,{(B_{i})}_{i\in I}])-{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A\bigcap B,{{\Big (}A_{i}\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/153ad47cd1c3c9f4166f4d27e68931b1d62dac5b)
![{\displaystyle \displaystyle {{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcup B,{{\Big (}A_{i}\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fee038ab688c0572885fb0c7dfb4e60e728955f)
![{\displaystyle \displaystyle {{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcap B,{{\Big (}A_{i}\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b993368881dc80fa624348813c38bae240cbeaf4)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\Big )}={card}_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}])+{card}_{Q,{\cal {R}}}([B,{(B_{i})}_{i\in I}])-{card}_{Q,{\cal {R}}}{\Big (}{\Big [}A,{(A_{i})}_{i\in I}{\Big ]}\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/595bdb116f94ac5a19375c32595c9e508bad7edd)
![{\displaystyle \forall [A,{(A)}_{i\in I}],[B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/436b500c4acd2ecf888a24e3d7509a6f7ad9b60e)
![{\displaystyle A\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A,{(A)}_{i\in I}{\Big ]}\bigcup {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcup B,{{\Big (}A\bigcup B_{i}{\Big )}}_{i\in I}{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b854a886c216a4498f2c9363d4d750715e8ae690)
![{\displaystyle A\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A,{(A)}_{i\in I}{\Big ]}\bigcap {\Big [}B,{(B_{i})}_{i\in I}{\Big ]}{\underset {d{\acute {e}}f}{=}}{\Big [}A\bigcap B,{{\Big (}A\bigcap B_{i}{\Big )}}_{i\in I}{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a64a6febcb633f14f19d5426d7ed29de2b5715e)
![{\displaystyle \displaystyle {\forall {(x_{m})}_{m\in \mathbb {N} }\subset \mathbb {R} ,\,\,convergente\,\,dans\,\,\mathbb {R} ,\,\,\lim _{m\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,x_{m}])={card}_{Q,{\cal {R}}}(\lim _{m\rightarrow \sup(\mathbb {N} )}[0,x_{m}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb2588c8fb364f36ce96fe4f1399c9f0507e849)
![{\displaystyle \displaystyle {\forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n-k}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f30aa567be9824a718dbeea7a73861a0e05993)
![{\displaystyle \displaystyle {\forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{k}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea54515f593a181005b9335c23b28f37b57ba77)
![{\displaystyle \displaystyle {[A\times B,{(A_{i}\times B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d55df2b36fa61bc31aac82abf21b0a79c92f2555)
![{\displaystyle \displaystyle {\forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n-k}),{PV}(\mathbb {R} ^{n-k}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbf4a5a4f36da2e4f592d7b112375d92f5028ba)
![{\displaystyle \displaystyle {\forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{k}),{PV}(\mathbb {R} ^{k}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb31a87fa7014e796590a542a4be5078ace41b1)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([A\times B,{(A_{i}\times B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n}}([A\times B,{(A_{i}\times B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n-k}}([A,{(A_{i})}_{i\in I}])\,\,{card}_{Q,{\cal {R}}_{k}}([B,{(B_{i})}_{i\in I}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e503e144aa85d79eb655b2a35ece25cf202579d5)
![{\displaystyle [A,{(A_{i})}_{i\in I}]\times [B,{(B_{i})}_{i\in I}]{\underset {d{\acute {e}}f}{=}}[A\times B,{(A_{i}\times B_{i})}_{i\in I}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f91c8180827c92bde1621bd8d4f0128897bbbc3d)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}([A,{(A_{i})}_{i\in I}]\times [B,{(B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n}}([A,{(A_{i})}_{i\in I}]\times [B,{(B_{i})}_{i\in I}])={card}_{Q,{\cal {R}}_{n-k}}([A,{(A_{i})}_{i\in I}])\,\,{card}_{Q,{\cal {R}}_{k}}([B,{(B_{i})}_{i\in I}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3795b09678c41e05465e228dabfd7f9f0eaa65f5)
![{\displaystyle \displaystyle {{\Big (}[A,{(A_{i})}_{i\in I}]\subset [B,{(B_{i})}_{i\in I}]{\Big )}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,(A\subset B\,\,et\,\,\forall i\in I,\,\,A_{i}\subset B_{i})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe83bf8e191bd7b2f05224a3c9b655b70fe417b)
![{\displaystyle \displaystyle {{\bigg (}[A,{(A_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[B,{(B_{i})}_{i\in I}]{\Big )}{\bigg )}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,{\bigg (}A\in {\mathcal {P}}(B)\,\,et\,\,{(A_{i})}_{i\in I}\in \prod _{i\in I}{\mathcal {P}}(B_{i}){\bigg )}\,\,{\underset {d{\acute {e}}f}{\Leftrightarrow }}\,\,{\Big (}A\in {\mathcal {P}}(B)\,\,et\,\,\forall i\in I,\,\,A_{i}\in {\mathcal {P}}(B_{i}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d981d28ff928abb5237bb4c96fee17e2c17e6d14)
![{\displaystyle \forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64f94b5d5083290594f94ea9b51b5d6681501c3)
![{\displaystyle [A\setminus B,{(A_{i}\setminus B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/327ae631721459185a4bed876afb75f5cf7a2caa)
![{\displaystyle \forall [A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52f13bb6b65c06fe7a1c4734df291f8ddbce6872)
![{\displaystyle \forall [B,{(B_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV2}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B_{i})}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2deaace4ea1107d7d458bc91acaa3c058fad691)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A\setminus B,{(A_{i}\setminus B_{i})}_{i\in I}])={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B_{i})}_{i\in I}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a22a62a681b6efa115194c968d4b873ea428ceae)
![{\displaystyle [A,{(A_{i})}_{i\in I}]\setminus [B,{(B_{i})}_{i\in I}]{\underset {d{\acute {e}}f}{=}}[A\setminus B,{(A_{i}\setminus B_{i})}_{i\in I}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5664a328b083b28b5c156d0e43a041e6c490804)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]\setminus [B,{(B_{i})}_{i\in I}])={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B_{i})}_{i\in I}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c262bcd2b14c3810f48bbde6678173da8282925)
![{\displaystyle \forall [B,{(B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B)}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51311578e7eda1cc9a9fca8685acbb911808465)
![{\displaystyle [A\setminus B,{(A_{i}\setminus B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0719393e2771c0e00b30f68b874198daa3bca5)
![{\displaystyle \forall [B,{(B)}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{PV}(\mathbb {R} ^{n}),{PV}(\mathbb {R} ^{n}){\Big )},\,\,[B,{(B)}_{i\in I}]\in {\mathcal {P}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70e8a088a843ed44bf7be385f6334b205a3053b3)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A\setminus B,{(A_{i}\setminus B)]}_{i\in I}])={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B)}_{i\in I}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8108200f733d3dc3eeb6937ff75a96abbda101c5)
![{\displaystyle [A,{(A_{i})}_{i\in I}]\setminus B{\underset {d{\acute {e}}f}{=}}[A,{(A_{i})}_{i\in I}]\setminus [B,{(B)}_{i\in I}]{\underset {d{\acute {e}}f}{=}}[A\setminus B,{(A_{i}\setminus B)]}_{i\in I}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14f8833b436a7e5420b671884b5fae83aa9fccb0)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]\setminus B)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b6215a551174c21f1aff39a210d25e9b593a78)
![{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]\setminus [B,{(B)}_{i\in I}])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa084e1c541d60b3ad56b28b78c1b133713d6d2b)
![{\displaystyle ={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}([B,{(B)}_{i\in I}])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15eac5d31ccc0a2f784d4fea9ef6e849980a6a90)
![{\displaystyle ={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}])-{card}_{Q,{\mathcal {R}}}(B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2956cff26cafda019e0f6841d336a7131cc3b2d)
![{\displaystyle \exists {(A_{i})}_{i\in I}\in {\Big \{}{(A_{i})}_{i\in I}\in {\mathcal {F}}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}\,\,{\Big |}\,\,[A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}{\Big \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65212c1cd65f72435e181d1b45e67d73ebf494a0)
![{\displaystyle {card}_{Q,{\mathcal {R}}}(A)={card}_{Q,{\mathcal {R}}}{\Big (}[A,{(A_{i})}_{i\in I}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63bb56f8fc3c69476736a6232b5c228a820f05f1)
![{\displaystyle [A,{(A_{i})}_{i\in I}]=[A,{(A)}_{i\in I}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f25569eb6cc1e885d1b9a2effc890f43365968fe)
![{\displaystyle \displaystyle {{{\mathcal {P}}lafonnements}_{normaux}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}{\underset {d{\acute {e}}f}{=}}{\Big \{}[A,{(A_{i})}_{i\in I}]\in {{\mathcal {P}}lafonnements}{\Big (}I,{\mathcal {P}}(\mathbb {R} ^{n}){\Big )}\,\,{\Big |}\,\,{card}_{Q,{\mathcal {R}}}(A)={card}_{Q,{\mathcal {R}}}([A,{(A_{i})}_{i\in I}]){\Big \}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8756c2b51ce626a71c75a8db400121448c48cb72)
![{\displaystyle R_{1,+}={\Big [}\mathbb {R} _{+},{([0,r])}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af7875c2f7130524ded3f05e192b3a3e9c42250f)
![{\displaystyle R_{2,+}={\Big [}\mathbb {R} _{+},{([0,r[)}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a78c3fc29fe631426173c4edbae58b07489fa27)
![{\displaystyle R_{1}={\Big [}\mathbb {R} _{+},{([-r,r])}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/636f9efe8eb802e5abfaf5607ec35f810fc4991b)
![{\displaystyle R_{2}={\Big [}\mathbb {R} _{+},{(]-r,r[)}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2053d12518ba1953677753e05d5c50c852f4d875)
![{\displaystyle \displaystyle {N_{1}={\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6d181e620019ffa3e577122526b8445448c708)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{2,+})={card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,p[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0a88690832afbd2416da47672db3da2a337b8e)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[i-1,i[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[)=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([i-1,i[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3dcd625481c4cad23e1aa17471fa0fcdecc3c8c)
![{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[)={card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c05bb605a6b524d9e5f0beefc3d487e8ecbee39)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca72d9423a520f8d600b4e5e016f0e48c775000)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,2p[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,2p[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19e0ebdaea99980035d8ea951337ffff160123f6)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[2i-2,2i[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([2i-2,2i[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47634f38c790b2b6a74888d66b0d3c0d060c26b4)
![{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([2i-2,2i[)=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,2[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5600334966f2164b69348111e78e739fe287de25)
![{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}2\,\,{card}_{Q,{\cal {R}}}([0,1[)=2\,\,{card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc011f6634e0465a75c5717d11733eb9d07ccc6)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08f317e21fcd263275dc688b3438d8c4fd571aad)
![{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}\neq {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e967278e434ee245cbdf027f7e6a32c65e83aca1)
![{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1be175a42ba20665e47482da66479a752185e58)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{1,+})={card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,p]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p])=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p[\bigsqcup \{p\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bed4b91a689a64e6f18abfe64b3fc8779f3ac35)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}\left(\left(\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[i-1,i[\right)\bigsqcup \{p\}\right)=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+{card}_{Q,{\cal {R}}}(\{p\}){\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c980fbdb9d0c0b846bcb839f2e229bf530478aeb)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1{\bigg )}={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98a5c94550aca6e62590e950ed9ad58f61971042)
![{\displaystyle \displaystyle {={\bigg (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[){\bigg )}+1={card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5238045fd757dd8bb3eb92719815b718e3196f)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d20495d1a8217d8164f4d14185bd24f227f403)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,2p]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,2p])=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,2p[\bigsqcup \{2p\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b6f353826391f4ad9b98bbd4ac895eb8ad13da)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}\left(\left(\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[2i-2,2i[\right)\bigsqcup \{2p\}\right)=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([2i-2,2i[){\Big )}+{card}_{Q,{\cal {R}}}(\{2p\}){\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e43efd53720faef104b825b2a80f2dc30502f461)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([2i-2,2i[){\Big )}+1{\bigg )}={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([2i-2,2i[){\Big )}+1={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,2[){\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae7b82b5b829088ae07fc2328b85c48201d9515)
![{\displaystyle \displaystyle {={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}2\,\,{card}_{Q,{\cal {R}}}([0,1[){\Big )}+1=2\,\,{card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4968b2bc4c7a9174d0b5a3f9fdfcf7a730dbc718)
![{\displaystyle \displaystyle {=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2221f8f24b623be363e366e8a9007ccc1296a8)
![{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}\neq {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c23fb4976d87f1ddd9d5c40f293d5fc6e61a32c)
![{\displaystyle {card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,2p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}=2\,\,{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2459cf9a42d6a4c68fb492afd045b29c19aba0)
![{\displaystyle {([0,p])}_{p\in \mathbb {N} }\neq {([0,2p])}_{p\in \mathbb {N} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5307e4b26e2f81e290ff807cf306f6ed899bbe39)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,f(p)[)}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,f(p)[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,f(p)[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce0b84c506e92989e19694ab9c01b7330636d7c)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[f(i-1),f(i)[{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([f(i-1),f(i)[)=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([0,f(i)-f(i-1)[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47b121fb3c965858b3ef38bedc793cc3faf5a3a1)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13e13775f84eb82591f4c1910aa44d52961ab308)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c56d78c068ce9712cad68b1649d8a85c3d7dc7)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f{\Bigg (}{{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {N} ,(\mathbb {N} \bigcap [0,p])_{p\in \mathbb {N} }{\bigg ]}}^{*}{\Bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81caa6aa8743e90b122871a1a4ea5ffb165b0756)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,f(p)])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,f(p)]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,f(p)])=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,f(p)[\bigsqcup \{f(p)\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13ecf1909759d47ea67f52513899bef7a1af5931)
![{\displaystyle \displaystyle {=\lim _{p\in \mathbb {N} ,p\rightarrow +\infty }{\bigg (}{card}_{Q,{\cal {R}}}{\Big (}\bigsqcup _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}[f(i-1),f(i)[{\Big )}+{card}_{Q,{\cal {R}}}(\{f(p)\}){\bigg )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([f(i-1),f(i)[){\Big )}+1{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0933c9a391f21d72d78aa09d2fb33efb5ce5622c)
![{\displaystyle \displaystyle {={\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([f(i-1),f(i)[){\bigg )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c00a5dded6bbea05d3eb62ce276c443b4eefacb1)
![{\displaystyle \displaystyle {={\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([0,f(i)-f(i-1)[){\bigg )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/221efd747b09eb09fde45b3686d4c8bd04207845)
![{\displaystyle \displaystyle {={\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}\,\,{card}_{Q,{\cal {R}}}([0,1[){\bigg )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68900cdf5ceb8684f2ed6628892eb5292b4c690e)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{\Big (}f(i)-f(i-1){\Big )}{\bigg )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ade15c40e7b4e2677ebb8ba5a7f963c7dbe2dce)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}([0,1[)\,\,f{\Bigg (}{{card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {N} ,(\mathbb {N} \bigcap [0,p])_{p\in \mathbb {N} }{\bigg ]}}^{*}{\Bigg )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8adbd1f0fecd142b5e3579e6064613e1f50f9beb)
![{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}1=\sum _{\displaystyle {i\in \lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}}}1=\sum _{\displaystyle {i\in {\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}(\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}}}1=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af4dc1edd833006b9f9ba339f8e2438435683ca1)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}={card}_{Q,{\cal {R}}}(N_{1}^{*})=\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07ef65a3c06c6827a7c74fd31bf4554bb690a41)
![{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}={card}_{Q,{\cal {R}}}{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}{\bigg )}={card}_{Q,{\cal {R}}}{\bigg (}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}(\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bca394b5deafae93c9e1fa98dea17546717fdee)
![{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)=f(\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p)=f{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}\sum _{i\in {\Big (}(\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}}1{\bigg )}=f{\bigg (}\sum _{\displaystyle {i\in \lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}}}1{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83030e4f64f787bd9915acf09d5a368e040e399d)
![{\displaystyle \displaystyle {=f{\bigg (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\bigg )}=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}{\Bigg )}=f{\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*}){\Big )}=f{\Big (}\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4a9eae50b16d5182f399770c6170c8c06ec874)
![{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}f(p)=f(\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}p)=f{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}{\bigg )}=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}(\mathbb {N} \bigcap [0,p])\setminus \{0\}{\Big )}{\bigg )}{\Bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a089d99d71709735d7bcdbaaf3249a304a11b6)
![{\displaystyle \displaystyle {=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\Big (}\mathbb {N} \bigcap [0,p]){\Big )}\setminus \{0\}{\bigg )}{\Bigg )}=f{\Bigg (}{card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}{\Bigg )}=f{\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*}){\Big )}=f{\Big (}\sup _{non\,\,classique,\,\,{\mathcal {R}}}(N_{1}){\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc1621be550150a7e033ac8b718bb96ebae7930)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}(R_{1,+})={card}_{Q,{\cal {R}}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}}{\Big (}\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}[0,p]{\Big )}=\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}}([0,p])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a07f832cedf2f9b8a2d26b911957433b358e13)
![{\displaystyle \displaystyle {\lim _{p\in \mathbb {N} ,p\rightarrow \sup(\mathbb {N} )}{\bigg (}{\Big (}\sum _{i\in (\mathbb {N} \bigcap [0,p])\setminus \{0\}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1{\bigg )}={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([i-1,i[){\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4f1069c9f9385832534daa7ff583dfdbf60576)
![{\displaystyle \displaystyle {={\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[){\Big )}+1={card}_{Q,{\cal {R}}}([0,1[)\,\,{\Big (}\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1{\Big )}+1={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9f72124f784c8efa5a14465167c03e7c3269b5)
![{\displaystyle \displaystyle {=\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}{card}_{Q,{\cal {R}}}([0,1[)={card}_{Q,{\cal {R}}}([0,1[)\sum _{i\in {{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}}1={card}_{Q,{\cal {R}}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{*}{\bigg )}\,\,{card}_{Q,{\cal {R}}}([0,1[)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a42674bdfaaf3cd8f77a0ff4e027213c39c2a6b3)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(\mathbb {N} \bigcap [0,p])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c490504a4d8b6e53d2f44d2ff727cd135141f3a)
![{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,p]\!]\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bae6c734a9a296d2d1c0aeeef806134ce08a479b)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(\{[\![0,{\underset {p\in \mathbb {N} ,\,\,p\to \sup(\mathbb {N} )}{{\text{lim}}_{classique}}}p]\!]\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d548addffc3a8bdefc8e6facbcebde8da473841c)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}(\{[\![0,\sup(\mathbb {N} )]\!]\})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3365d2fa6160408c6b037a85150880c49c49131e)
![{\displaystyle \displaystyle {{card}_{Q,{\mathcal {R}}}([\![0,\displaystyle {{\underset {p\in \mathbb {N} \bigsqcup \{\sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )\},\,\,p\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}{{\text{lim}}_{classique}}}p}]\!])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea7126d11a14828430c1b03b3eb7b29727119bc)
![{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,\sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )]\!])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23ab907b17e080dbd6c4d66cbeac7d7716ea4592)
![{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,{\displaystyle {\lim _{p\in \mathbb {N} ,\,\,p\rightarrow \sup _{non\,\,classique,\,\,{\mathcal {R}}}(\mathbb {N} )}p}]\!])}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07cd68bd6ca0e48b8ea38ffb07f74a6f421f18f5)
![{\displaystyle \displaystyle {={card}_{Q,{\mathcal {R}}}([\![0,{\displaystyle {\lim _{p\in \mathbb {N} ,\,\,p\rightarrow \sup(\mathbb {N} )}p}]\!])}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd76b8863b909cde661b1c70143746fa5155b7d9)
![{\displaystyle R_{2}={\Big [}\mathbb {R} ,{(]-r,r[)}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e72f5e8279b5367e79d04beee448f12a956533)
![{\displaystyle R_{2,-}={\Big [}\mathbb {R} _{-},{(]-r,0])}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d62ac4cc052f363f5c17198b3048280e537701a6)
![{\displaystyle \displaystyle {-N_{1}={\Big [}-\mathbb {N} ,{(-\mathbb {N} \bigcap [-p,0])}_{p\in \mathbb {N} }{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4aaf47a81991b635b068a1391d1384fc99073b1)
![{\displaystyle \displaystyle {Z_{1}={\Big [}\mathbb {Z} ,{(\mathbb {Z} \bigcap [-p,p])}_{p\in \mathbb {N} }{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20a1f9e8ca0c42210db07bfb0867c06c18a5a800)
![{\displaystyle {card}_{Q,{\cal {R}}}(R_{2,+}^{*})={card}_{Q,{\cal {R}}}(R_{2,+})-1={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-1={card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,1[)+1{\Big )}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/550678caf9a06b13c0e98d61f19756fc9d77c5c4)
![{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(R_{2,+}^{*})+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(N_{1}^{*})={card}_{Q,{\cal {R}}}(N_{1})-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a21909e6872177085d80b82353c99c7cd33725f)
![{\displaystyle =2\,\,{card}_{Q,{\cal {R}}}(N_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-1={card}_{Q,{\cal {R}}}(Z_{1}^{*})\,\,{card}_{Q,{\cal {R}}}([0,1[)-1={card}_{Q,{\cal {R}}}(Z_{1}^{*})\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,1[)+1{\Big )}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c40d572a873d4d384b37adfdd8886bf275ea44)
![{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}(R_{2})+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(Z_{1}^{*})={card}_{Q,{\cal {R}}}(Z_{1})-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00e545283b32db0dd4b2e5e37e1fdf151b12bcc3)
![{\displaystyle {R}_{2,+}={\Big [}{\mathbb {R} }_{+},{([0,r[)}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c1721ee844e0d0a1cbcffac63789fef704a653f)
![{\displaystyle N_{1}={\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/256e0bdd6de95df4c8139490f8cf1bdc60c2bdf0)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])={card}_{Q,{\cal {R}}}(]a,b])+{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,b])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a4acd7e19aaf02f1ae420cc418f6cecf15ec51)
![{\displaystyle {card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c155bda98f162677ddd87cb5b1f4ac1bfee7db1)
![{\displaystyle ={\Big (}{card}_{Q,{\cal {R}}}(N_{1}^{*})-a{\Big )}\,\,{\Big (}{card}_{Q,{\cal {R}}}(]0,1[)+1{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72e62ca74d1059e9304511f8d7240b38b6650b4e)
![{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(N_{1}^{*})-a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/701acbe0617f6df5ae89df2a52fe849527fd0a14)
![{\displaystyle {card}_{Q,{\cal {R}}}({R}_{2,+}\bigcup ]a,0[)+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c134566002542cae5b6c620a85fc7a280d01941)
![{\displaystyle \displaystyle {{\frac {{card}_{Q,{\cal {R}}}({R}_{2,+}\bigcup ]a,0[)+1}{{card}_{Q,{\cal {R}}}(]0,1[)+1}}={card}_{Q,{\cal {R}}}(N_{1}^{*})-a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37b65fb3627e617bb8b5b7ed9a86ce31d87cc22f)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a])={card}_{Q,{\cal {R}}}({R}_{2,-}\setminus [-a,0])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/252c4712eb67447e126497f8ced8f3a7925914af)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}}({R}_{2,+}\setminus [0,a[)={card}_{Q,{\cal {R}}}({R}_{2,-}\setminus ]-a,0])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ebf6984a6ecd6821962d2f55b578bcba20a6fa)
![{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a998454dd28d211c375a26de936d3d90772a0fc)
![{\displaystyle \displaystyle {{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}[-r,r]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d43e2f4a93d95e0713df73ef6d408f7e98dc7b18)
![{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}={{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}}^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3181e6d89e8a9639505df1b748f468dda79a2a)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}={{\Bigg (}{card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}{\Bigg )}}^{2}={card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9087b7686859b1a23c331f1d5c100939a890b0a4)
![{\displaystyle \displaystyle {{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d532eef69a3d1fcd528b5a4a3a3b33851f0d863)
![{\displaystyle \displaystyle {{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}=\lim _{r\in \mathbb {N} ,r\rightarrow +\infty }{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a921efe0a4a30242c285ec5f43fbe5b0a3e33c17)
![{\displaystyle \displaystyle {\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}={\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6f3bb54e3b602f1343f466595fe693ee294ea2)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\Bigg (}{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}{\Bigg )}={card}_{Q,{\cal {R}}_{2}}{\Big (}\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cdb2856a9444236166f16604e41370a897e107)
![{\displaystyle {card}_{Q,{\cal {R}}_{1}}([0,1[)={card}_{Q,{\cal {R}}_{1}}([0,1])-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6ce841958cef16f8a2b71424c304798a2825fd)
![{\displaystyle {card}_{Q,{\cal {R}}_{2}}({[0,1[}^{2})={card}_{Q,{\cal {R}}_{1}}^{2}([0,1[)={{\Big (}{card}_{Q,{\cal {R}}_{1}}([0,1])-1{\Big )}}^{2}={card}_{Q,{\cal {R}}_{1}}^{2}([0,1])-2\,\,{card}_{Q,{\cal {R}}_{1}}([0,1])+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad81cc3cfb5aa96b35ccd1feb0d2b9b10833a5f)
![{\displaystyle {card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},1)}}{\Big )}=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}([0,1])-\pi \,\,{card}_{Q,{\cal {R}}_{1}}([0,1])+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/081f1eed5e8b84eb005e002cd5c1eb3baff6f6ec)
![{\displaystyle \forall r\in \mathbb {N} ,\,\,{card}_{Q,{\cal {R}}_{2}}{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}([0,r])-\pi \,\,{card}_{Q,{\cal {R}}_{1}}([0,r])+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/180c9d0c507829b377efb0b9880e172d28efe747)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\Bigg (}{\bigg [}\mathbb {R} ^{2},{{\Big (}{\overline {B_{\mathbb {R} ^{2}}(O_{2},r)}}{\Big )}}_{r\in \mathbb {N} }{\bigg ]}{\Bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7220f1b20d386aa3bc707d00ff00f413323cdba3)
![{\displaystyle \displaystyle {=\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{\Big (}\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}([0,r])-\pi \,\,{card}_{Q,{\cal {R}}_{1}}([0,r])+1{\Big )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3853ec84a9b2b25d5881d70b38eb5259366b7a62)
![{\displaystyle \displaystyle {=\pi \,\,\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}_{1}}^{2}([0,r])-\pi \,\,\lim _{r\in \mathbb {R} _{+},r\rightarrow \sup(\mathbb {N} )}{card}_{Q,{\cal {R}}_{1}}([0,r])+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/402d622edef8ca87022319176f942674e8415247)
![{\displaystyle \displaystyle {=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}{\Big (}\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}[0,r]{\Big )}-\pi \,\,{card}_{Q,{\cal {R}}_{1}}{\Big (}\lim _{r\in \mathbb {N} ,r\rightarrow \sup(\mathbb {N} )}[0,r]{\Big )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/824af7d5191063086c3b366e675cc37d20a0ee69)
![{\displaystyle \displaystyle {=\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}-\pi \,\,{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} _{+},{([0,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d859f07bd85ce3a84179e83790ae2d5349a571d6)
![{\displaystyle \displaystyle {={\frac {1}{4}}\pi \,\,{{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}+1{\Bigg )}}^{2}-{\frac {1}{2}}\pi \,\,{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}+1{\Bigg )}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b573e4b40562ceac8678340ec5709a33394a5a71)
![{\displaystyle \displaystyle {={\frac {1}{4}}\pi \,\,{card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}-{\frac {1}{4}}\pi +1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/667fd86c044033efa4fa86366efa44add408eb1f)
![{\displaystyle \displaystyle {\neq {card}_{Q,{\cal {R}}_{1}}^{2}{\bigg (}{\Big [}\mathbb {R} ,{([-r,r])}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b87ac282be3c4e315363424562a4261be493f4b)
![{\displaystyle \displaystyle {={card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c87aecfc96ccbf492d226e2f94640a322767e9)
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
![{\displaystyle [0,10]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af219c8f0e8e0381e7b35e63dd7093e86d83a540)
![{\displaystyle {\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e607cf38a03e839f094459c28ad69a532a16b70)
![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\bigg (}{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}={{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}{\Bigg )}}^{2}>{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c214ea89fba1fa694854e6b588b6096b2063cf4f)
![{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}={{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}=\bigsqcup _{x\in {\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}{\bigg \{}(x,y)\in {{\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}}^{2}{\bigg |}y\in {\Big [}\mathbb {R} ,{({[-r,r]})}_{r\in \mathbb {N} }{\Big ]}{\bigg \}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9da5a8593e6d0e2ad3adc7660767ff527213e77)
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![{\displaystyle \displaystyle {{\Big [}\mathbb {R} ^{2},{({[-r,r]}^{2})}_{r\in \mathbb {N} }{\Big ]}=R_{1}^{2}=\bigsqcup _{x\in R_{1}}\{(x,y)\in R_{1}^{2}|y\in R_{1}\}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd200f71c46f8a684fe5bea90b1a2a176598b1e5)
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![{\displaystyle \displaystyle {{card}_{Q,{\cal {R}}_{2}}{\Bigg (}{\bigg [}\mathbb {N} ^{2},{{(\mathbb {N} \bigcap [0,p])}^{2}}_{p\in \mathbb {N} }{\bigg ]}{\Bigg )}={card}_{Q,{\cal {R}}_{2}}{\bigg (}{{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}}^{2}{\bigg )}={{\Bigg (}{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}{\Bigg )}}^{2}>{card}_{Q,{\cal {R}}_{1}}{\bigg (}{\Big [}\mathbb {N} ,{(\mathbb {N} \bigcap [0,p])}_{p\in \mathbb {N} }{\Big ]}{\bigg )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed4ad06a3c8e044c207887c44f5283339c7c17b)
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