Definition 5.5.1 (Outer Measure).label Let $X$ be a set and $\mu^{*}:2^{X}\to[0,\infty]$ then $\mu^{*}$ is an outer measure if:

1. (OM1)

   $\mu^{*}(\emp)=0$

2. (OM2)

   For any $E,F\suf X$ with $E\suf F,\mu^{*}(E)\leq \mu^{*}(F)$

3. (OM3)

   For any $\curl{E_n}_{n\in\N^+}\suf X$

   \begin{align*}\mu^{*}\parens{\cups{}{n\in\N^+}E_n}\leq \sums{}{n\in\N^+}\mu^{*}(E_{n})\end{align*}