Carathéodory construction

A method that turns an outer measure into a measure by selecting Carathéodory measurable sets.

Carathéodory construction

Carathéodory construction: Let $\mu^*$ be an outer measure on a set $X$ . Define

\mathcal{M} =\Bigl\{E\subseteq X : \mu^*(A)=\mu^*(A\cap E)+\mu^*(A\setminus E)\ \text{for every } A\subseteq X\Bigr\}.

Then $\mathcal{M}$ is a sigma-algebra on $X$ , and the restriction $\mu:=\mu^*|_{\mathcal{M}}$ is a measure on $\mathcal{M}$ . The sets in $\mathcal{M}$ are exactly the Carathéodory measurable sets associated to $\mu^*$ .

This construction is a standard way to build a measurable space and a measure from an outer measure; in particular it underlies the definition of Lebesgue measure on $\mathbb{R}^n$ .