Definition (Outer measure)

The map $\mu^{*}:2^{X}\to\overline{\mathbb{R}}_{+}$ is an outer measure if:

1. $\mu^{*}(\emptyset)=0$
2. Monotonicity: $A\subset B\implies \mu^{*}(A)\le\mu^{*}(B)$
3. Countable Subadditivity: $$A_{1},A_{2},\dots\in2^{X}\implies \mu^{*}\left( \bigcup_{n=1}^\infty A_{n} \right)\le\sum_{n=1}^{\infty}\mu^{*}(A_{n})$$