Outer measure

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

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

Definition.

A set $A\subset X$ is $\mu ^{*}$-measurable if $\mu ^{*}(E)=\mu ^{*}(E\cap A)+\mu ^{*}(E\setminus A)$ for all $E\subset X$.