About the Caratheodory class.

Question

Let $X$ a set, and $\mathcal{P}(X)$ the class of its subset's. Let $\mathcal{A}\subset \mathcal{P}(X)$, we call a map $L: \mathcal{P}(X)\to[0, \infty]$ $\mathcal{A}$-regular if for any $S\subset X$ we have $L(S)= inf_{S\subset A\in \mathcal{A}} L(A)$.

A map $L: \mathcal{P}(X)\to[0, \infty]$ is called a outer-measure if

$L(\emptyset)=0$, $A\subset B\Rightarrow L(A)\leq L(B)$, $L(\bigcup_{n\in \mathbb{N}} A_n) \leq \sum_{n\in \mathbb{N}} L(A_n)$

Form a outer-measure $L$ we have the Carathéodory's construction associated measure: $(X, \mathcal{A}_L, \mu_L)$ where $\mathcal{A}_L$ is the class of subset $A\subset X$ such that $\forall S\subset X: L(S)=L(S\setminus A)+ L(S\cap A)$, and $\mu_L$ is the restriction of $L$.

Let $\mu: \mathcal{A}\to[0, \infty[$ a real finite ($\sigma$-additive) measure on a a $\sigma$-algebra $\mathcal{A}\subset \mathcal{P}(X)$, we can define the associated outer-measure $\mu^\ast: \mathcal{P}(X)\to[0, \infty]$ as $\mu^\ast(S):= inf_{S\subset A\in \mathcal{A}} \mu(A)$. It is well know that if $(X, \mathcal{A'}, \mu')$ is its Carathéodory' constructiuon then $\mathcal{A}\subset \mathcal{A'}$ and $\mu$ is the restriction of $\mu'$, and $\mu^\ast$ is $\mathcal{A}$-regular and then $\mathcal{A'}$-regular.

THen I ask: If I have a outer measure $L$ as above

and its Carathéodory's construction $(X, \mathcal{A}_L, \mu_L)$

is $L$ a $\mathcal{A}_L$-regular map?

Motive of the question: I try to study a (a adjoint functors pair?) correspondence between the measure and the outer measure on a set.

Answer 1

Given a set $X$ and outer measure $L:\mathcal{P}(X)\to [0,\infty]$, is $L$ an $\mathcal{A}_L$-regular map? The answer is No. Consider the following example:

Let $X = \mathbb{N}$ and define $L:\mathcal{P}(\mathbb{N}) \to [0,\infty]$ by $L(\emptyset) = 0$ and $L(F) = 1$ for $F\subseteq \mathbb{N}$ finite, and $L(S) = \infty$ for $S\subseteq\mathbb{N}$ infinite.

It is easy to see that $L$ is an outer measure.

Observation: If $F\subseteq \mathbb{N}$ is non-empty and finite, then $F\notin \mathcal{A}_L$.

Proof : Let $F\subseteq \mathbb{N}$ be a non-empty finite set. Then let $F^*:= F \cup \{\max(F) + 1\}$. So $L(F^*) = 1$ but $L(F^*\setminus F) + L(F^*\cap F) = 2$.

Now consider the finite set $\{1\}$. We have $L(\{1\}) = 1$, but the observation above implies that for any $A\in \mathcal{A}_L$ with $\{1\} \subseteq A$ we have that $A$ is infinite, therefore $L(A) = \infty$. So $\inf\{L(A): \{1\} \subseteq A\}=\infty$, therefore $1 = L(1) \neq \inf\{L(A): \{1\} \subseteq A\}$. So $L$ is not $\mathcal{A}_L$-regular.