Outer measure

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 $(A_n)_{n\ge 1}$,

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 .