Let $X$ be a set and let $\mu^{*}$ be an \reftext{def:outer-measure-2026a}{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 \reftext{def:sigma-algebra-measurable-space-2026a}{$\sigma$-algebra} on $X$;
2. the restriction of $\mu^{*}$ to $\mathcal{M}$ is a \reftext{def:measure-measure-space-2026a}{measure} on $(X,\mathcal{M})$.