Outer Measure and Caratheodory Measurability

Let $X$ be a set. An outer measure on $X$ is a function $\mu^{*}$ from the family of all subsets of $X$ to $[0,\infty]$ (with the conventions of Measure, Measure Space, and Probability Measure) such that:

1. $\mu^{*}(\varnothing)=0$;
2. (monotonicity) if $A\subseteq B\subseteq X$ then $\mu^{*}(A)\le\mu^{*}(B)$;
3. (countable subadditivity) for every sequence $(A_m)_{m\in\mathbb{N}}$ of subsets of $X$,

with the sum as in Measure, Measure Space, and Probability Measure.

A subset $E\subseteq X$ is Carathéodory measurable with respect to $\mu^{*}$ if for every subset $A\subseteq X$,

where $A\setminus E$ is the complement of $E$ relative to $A$.