User:Nannou7/Caratheodory Dimension Structure

Caratheodory Dimension Structures form the basis for many aspects of modern dimension theory.

Formal Definition
Let $$X$$ be a set, $$\mathcal{F}$$ a collection of subsets of $$X$$, and $$\eta, \psi, \xi : \mathcal{F} \rightarrow [0, \infty )$$ be set functions satisfying the following conditions:
 * 1) $$ \empty \in \mathcal{F}, \eta ( \empty ) = 0, \psi ( \empty ) = 0 $$
 * 2) $$ \forall U \in \mathcal{F}, U \neq \empty \mbox{ we have that } \eta ( U ) > 0, \psi ( U ) > 0$$
 * 3) $$ \forall \delta > 0, \exists \varepsilon > 0 \mbox{ such that } \eta ( U ) \le \delta \forall U \in \mathcal{F} \mbox{ with } \psi ( U ) \le \varepsilon $$
 * 4) $$ \forall \varepsilon > 0, \exists \mbox{ finite or countable subcollection } \mathcal{G}  \subset \mathcal{F} \mbox{ covering } \mathcal{F} \mbox{, with } \psi ( \mathcal{G} ) := sup \{ \psi ( U ) : U \in \mathcal{G} \}  \le \varepsilon $$

If these hold, say $$\mathcal{F}, \xi, \eta, \psi$$ introduce a Caratheodory dimension structure or C-structure $$\tau$$ on $$X$$, and write $$\tau = ( \mathcal{F}, \xi, \eta, \psi )$$. Note especially that (almost) restriction at all is placed on $$\xi$$

Caratheodory Dimension
Given a set $$X$$ endowed with a C-structure as above, $$\alpha \in \mathbb{R}, \varepsilon > 0$$ and a set $$Z \subset X$$. Can define $$ M_C (Z, \alpha, \varepsilon) = \text{inf}_G \{ \sum_{U \in \mathcal{G}} \xi(U) \eta(U)^\alpha \}$$ where the infimum is over all countable subcollections $$\mathcal{G} \subset \mathcal{F}$$ covering Z, with $$\psi(G) \le \varepsilon $$.

$$M_C$$ is non-decreasing as $$\varepsilon$$ decreases. Therefore we can define:

$$m_C(Z, \alpha) = \lim_{\varepsilon \rightarrow 0}M_C(Z, \alpha, \varepsilon)$$

$$m_C( \cdot, \alpha )$$ is the $$\alpha$$-Caratheodory Outer measure

It can be shown that $$m_C(Z, \alpha_C)$$ can be $$0, \infty$$, or a finite positive number, and that the following is well defined.

The Caratheodory dimension of a set $$Z \subset X$$ is defined as:

$$\text{dim}_C Z = \text{inf}\{ \alpha : m_C(Z, \alpha) = 0 \} = \text{sup} \{ \alpha : m_C(Z, \alpha) = \infty \}$$

Hausdorff dimension, and Topological entropy can be defined using Caratheodory dimension by choosing a suitable C-structure.

Examples
A caratheodory dimension structure can be defined on $$\mathbb{R}^n$$ as follows: $$\mathcal{F}$$ is the collection of open sets, $$\forall U \in \mathcal{F}, \xi(U)=1, \eta(U)=\psi(U)= \text{diam}(U)$$