Definition:Outer Measure

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


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