Carathéodory construction

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

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 $\mathbb{R}^n$.