Definition 18.6.3 (Outer Measurable).label Let $X$ be a set, $\mu^{*}: 2^{X} \to [0, \infty]$ be an outer measure, and $E \subset X$, then $E$ is $\mu^{*}$-measurable if for any $F \subset X$,
\[\mu^{*}(F) = \mu^{*}(E \cap F) + \mu^{*}(E \setminus F)\]