Outer measure - Imported Knowlpedia Content

An outer measure on a set $X$ is a function $\mu^*:\mathcal P(X)\to[0,\infty]$ such that $\mu^*(\varnothing)=0$ , $\mu^*(A)\le \mu^*(B)$ whenever $A\subseteq B$ , and for every sequence $(A_n)_{n\ge 1}$ ,

\mu^*\!\left(\bigcup_{n=1}^\infty A_n\right)\le \sum_{n=1}^\infty \mu^*(A_n).

Outer measures live on the full power set and are used to define Carathéodory measurable sets. The Carathéodory construction turns an outer measure into a genuine measure on a sigma-algebra.

Examples

If $(X,\Sigma,\mu)$ is a measure space, then $\mu^*(A)=\inf\{\mu(B): B\in\Sigma,\ A\subseteq B\}$ defines an outer measure on $X$ .
Lebesgue outer measure on $\mathbb R^n$ is an outer measure built from coverings by rectangles and is the starting point for Lebesgue measure.