Outer measure

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

From the second and third conditions, we see that $A\subseteq \cup _{i=1}^{\infty }B_{i}$ implies $\mu ^{*}(A)\leq \sum _{i=1}^{\infty }\mu ^{*}(B_{i})$.

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