Definition:Outer Measure

Given a set $$X\ $$, an outer measure on $$X\ $$ is a function $$\mu^*:\mathcal P (X)\to \R$$ satisfying the following conditions:


 * $$\mu^*(A)\geq 0$$ for each $$A\in\mathcal P (X)$$,
 * $$\mu^*(\varnothing) = 0$$,
 * $$\mu^*(A) \leq \mu^*(B)$$ if $$A\subseteq B\in \mathcal P(X)$$ (i.e., $$\mu^*\ $$ is a monotonic function), and
 * $$\mu^*\Big(\bigcup_{i=1}^\infty A_i\Big) \leq \sum_{i=1}^\infty \mu^*(A_i)$$ for each countable collection of sets $$A_n\in\mathcal P(X)$$ (i.e., $$\mu^*\ $$ is countably subadditive).