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Caratheodory Dimension Structures form the basis for many aspects of modern dimension theory.
Formal Definition
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Let
X
{\displaystyle X}
F
{\displaystyle {\mathcal {F}}}
X
{\displaystyle X}
η
,
ψ
,
ξ
:
F
→
[
0
,
∞
)
{\displaystyle \eta ,\psi ,\xi :{\mathcal {F}}\rightarrow [0,\infty )}
∅
∈
F
,
η
(
∅
)
=
0
,
ψ
(
∅
)
=
0
{\displaystyle \emptyset \in {\mathcal {F}},\eta (\emptyset )=0,\psi (\emptyset )=0}
∀
U
∈
F
,
U
≠
∅
we have that
η
(
U
)
>
0
,
ψ
(
U
)
>
0
{\displaystyle \forall U\in {\mathcal {F}},U\neq \emptyset {\mbox{ we have that }}\eta (U)>0,\psi (U)>0}
∀
δ
>
0
,
∃
ε
>
0
such that
η
(
U
)
≤
δ
∀
U
∈
F
with
ψ
(
U
)
≤
ε
{\displaystyle \forall \delta >0,\exists \varepsilon >0{\mbox{ such that }}\eta (U)\leq \delta \forall U\in {\mathcal {F}}{\mbox{ with }}\psi (U)\leq \varepsilon }
∀
ε
>
0
,
∃
finite or countable subcollection
G
⊂
F
covering
F
, with
ψ
(
G
)
:=
s
u
p
{
ψ
(
U
)
:
U
∈
G
}
≤
ε
{\displaystyle \forall \varepsilon >0,\exists {\mbox{ finite or countable subcollection }}{\mathcal {G}}\subset {\mathcal {F}}{\mbox{ covering }}{\mathcal {F}}{\mbox{, with }}\psi ({\mathcal {G}}):=sup\{\psi (U):U\in {\mathcal {G}}\}\leq \varepsilon }
If these hold, say
F
,
ξ
,
η
,
ψ
{\displaystyle {\mathcal {F}},\xi ,\eta ,\psi }
τ
{\displaystyle \tau }
X
{\displaystyle X}
τ
=
(
F
,
ξ
,
η
,
ψ
)
{\displaystyle \tau =({\mathcal {F}},\xi ,\eta ,\psi )}
ξ
{\displaystyle \xi }
Caratheodory Dimension
edit
Given a set
X
{\displaystyle X}
α
∈
R
,
ε
>
0
{\displaystyle \alpha \in \mathbb {R} ,\varepsilon >0}
Z
⊂
X
{\displaystyle Z\subset X}
M
C
(
Z
,
α
,
ε
)
=
inf
G
{
∑
U
∈
G
ξ
(
U
)
η
(
U
)
α
}
{\displaystyle M_{C}(Z,\alpha ,\varepsilon )={\text{inf}}_{G}\{\sum _{U\in {\mathcal {G}}}\xi (U)\eta (U)^{\alpha }\}}
G
⊂
F
{\displaystyle {\mathcal {G}}\subset {\mathcal {F}}}
ψ
(
G
)
≤
ε
{\displaystyle \psi (G)\leq \varepsilon }
M
C
{\displaystyle M_{C}}
ε
{\displaystyle \varepsilon }
m
C
(
Z
,
α
)
=
lim
ε
→
0
M
C
(
Z
,
α
,
ε
)
{\displaystyle m_{C}(Z,\alpha )=\lim _{\varepsilon \rightarrow 0}M_{C}(Z,\alpha ,\varepsilon )}
m
C
(
⋅
,
α
)
{\displaystyle m_{C}(\cdot ,\alpha )}
α
{\displaystyle \alpha }
Outer measure
It can be shown that
m
C
(
Z
,
α
C
)
{\displaystyle m_{C}(Z,\alpha _{C})}
0
,
∞
{\displaystyle 0,\infty }
The Caratheodory dimension of a set
Z
⊂
X
{\displaystyle Z\subset X}
dim
C
Z
=
inf
{
α
:
m
C
(
Z
,
α
)
=
0
}
=
sup
{
α
:
m
C
(
Z
,
α
)
=
∞
}
{\displaystyle {\text{dim}}_{C}Z={\text{inf}}\{\alpha :m_{C}(Z,\alpha )=0\}={\text{sup}}\{\alpha :m_{C}(Z,\alpha )=\infty \}}
Hausdorff dimension , and Topological entropy can be defined using Caratheodory dimension by choosing a suitable C-structure.
Examples
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References
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