Outer measure

Let $X$ be a nonempty set. An outer measure on $X$ is a function $\mu ^{*}:2^{X}\to [0,\infty ]$ such that

• $\mu ^{*}(\emptyset )=0$
• $\mu ^{*}(A)\leq \mu ^{*}(B)$ if $A\subseteq B$
• $\mu *\left(\bigcup _{j=1}^{\infty }A_{j}\right)\leq \sum _{j=1}^{\infty }\mu ^{*}(A_{j}).$