Definition 18.6.1 (Outer Measure).label Let $X$ be a set and $\mu^{*}: 2^{X} \to [0, \infty]$, then $\mu^{*}$ is an outer measure if:

1. (M1)

   $\mu^{*}(\emptyset) = 0$.

2. (OM1)

   For any $E, F \subset X$ with $E \subset F$, $\mu^{*}(E) \le \mu^{*}(F)$.

3. (OM2)

   For any $\seq{E_n}\subset X$,

   \[\mu^{*}\paren{\bigcup_{n \in \natp}E_n}\le \sum_{n \in \natp}\mu^{*}(E_{n})\]