Caratheodory Extension Theorem

Let $X$ be a set and let $\mu^{*}$ be an outer measure on $X$. Let $\mathcal{M}$ be the family of all subsets of $X$ that are Carathéodory measurable with respect to $\mu^{*}$, in the sense of that definition. Then:

1. $\mathcal{M}$ is a $\sigma$-algebra on $X$;
2. the restriction of $\mu^{*}$ to $\mathcal{M}$ is a measure on $(X,\mathcal{M})$.