# Outer measure

Definition. Let ${\displaystyle X}$ be a nonempty set. An outer measure [1] on the set ${\displaystyle X}$ is a function ${\displaystyle \mu ^{*}:2^{X}\to [0,\infty ]}$ such that
• ${\displaystyle \mu ^{*}(\emptyset )=0}$,
• ${\displaystyle \mu ^{*}(A)\leq \mu ^{*}(B)}$ if ${\displaystyle A\subseteq B}$,
• ${\displaystyle \mu ^{*}\left(\bigcup \limits _{j=1}^{\infty }A_{j}\right)\leq \sum _{j=1}^{\infty }\mu ^{*}(A_{j}).}$

The second and third conditions in the definition of an outer measure are equivalent to the condition that ${\displaystyle A\subseteq \bigcup \limits _{i=1}^{\infty }B_{i}}$ implies ${\displaystyle \mu ^{*}(A)\leq \sum _{i=1}^{\infty }\mu ^{*}(B_{i})}$.

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

## Examples of Outer Measures

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

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

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

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

where ${\displaystyle F}$ is any right-continuous function [2].

Given a measure space ${\displaystyle (X,{\mathcal {M}},\mu )}$, one can always define an outer measure ${\displaystyle \mu ^{*}}$[3] by

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

## References

1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §1.4
2. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §1.5
3. Craig, Katy. MATH 201A HW 3. UC Santa Barbara, Fall 2020.