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

The second and third conditions in the definition of an outer measure are equivalent to the condition that $A\subseteq \bigcup \limits _{i=1}^{\infty }B_{i}$ 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 $E\subset X$.

Examples of Outer Measures

The standard example of an outer measure is the Lebesgue outer measure, defined on subsets of $\mathbb {R}$.

$\mu ^{*}(A)=\inf \left\{\sum _{i=1}^{\infty }|b_{i}-a_{i}|:A\subseteq \bigcup _{i=1}^{\infty }(a_{i},b_{i})\right\}.$

A near-generalization of the Lebesgue outer measure is given by

$\mu _{F}^{*}(A)=\inf \left\{\sum _{i=1}^{\infty }|F(b_{i})-F(a_{i})|:A\subseteq \bigcup _{i=1}^{\infty }(a_{i},b_{i}]\right\},$

where $F$ is any right-continuous function.

Given a measure space $(X,{\mathcal {M}},\mu )$, one can always define an outer measure $\mu ^{*}$ by

$\mu ^{*}(A)=\inf \left\{\mu (B):A\subseteq B,B\in {\mathcal {M}}\right\}.$