Let $\mu^{*}$ be an outer measure on $X$. Let $$\mathscr{M}=\{ A\subseteq X:A\text{ is }\mu^{*}\text{-measurable} \}$$then $\mathscr{M}$ is a σ-algebra, $\left.\mu^{*}\right|_{\mathscr{M}}$ is a measure, and $(X,\mathscr{M},\left.\mu^{*}\right|_{\mathscr{M}})$ is a complete measure space.