# Caratheodory and Riesz

This is related to a previous question.

Carathéodory's construction assigns to each outer measure $$\phi^+:\mathcal{P}(S)\to[0,\infty]$$ a $\sigma$-algebra $\Sigma$ where the restriction $\phi$ of $\phi^+$ is a measure. $\Sigma$ is easy to define: It is the collection of all those $X\subseteq S$ such that for any $A\subseteq S$, we have $$\phi^+(A)=\phi^+(A\cap X)+\phi^+(A\setminus X).$$

Riesz representation theorem also considers an outer measure and extracts from it a $\sigma$-algebra where the outer measure is a (nice, essentially regular) measure. In this case, $S$ is assumed to be Hausdorff and locally compact. We are given a positive linear functional $\Lambda\in C_c(S)$ (the collection of continuous complex-valued functions on $S$ with compact support). We start by defining $$\phi^+(V)=\sup\{\Lambda f\mid {\rm supp}(f)\subset V\}$$ for $V$ open, and then extend $\phi^+$ to all ${\mathcal P}(S)$ by $$\phi^+(E)=\inf\{\phi^+(V)\mid V\text{ is open, and }E\subseteq V\}.$$ This is an outer measure. The $\sigma$-algebra one associates to $\phi^+$ is $\mathcal{M}$, the collection of all $X\subseteq S$ such that for any compact $K\subseteq S$ we have that $\phi^+(X\cap K)<\infty$ and $$\phi^+(X\cap K)=\sup\{\phi^+(C)\mid C\text{ is compact, and }C\subseteq X\cap K\}.$$ Again, the restriction $\phi$ of $\phi^+$ to $\mathcal{M}$ is a measure.

Are there any reasonable circumstances under which we should expect that $\Sigma=\mathcal{M}$, or at least that one of the two containments holds?

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The answer is yes. The inclusion is easy in one direction and, in the other direction you can reduce to the case where $A$ is open and, then, to the case where $A$ is compact, so that the two definitions reduce to the same thing. –  George Lowther Jun 22 '12 at 14:38
Oh, you are right! Thanks. –  Bruce Jun 22 '12 at 16:42