Let $X$ be a set. An \textbf{outer measure} on $X$ is a function $\mu^{*}$ from the family of all subsets of $X$ to $[0,\infty]$ (with the conventions of \ref{def:measure-measure-space-2026a}) 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 \reftext{def:sequence-in-set-2026a}{sequence} $(A_m)_{m\in\mathbb{N}}$ of subsets of $X$,

with the sum as in \ref{def:measure-measure-space-2026a}.

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

where $A\setminus E$ is the \reftext{def:complement-subset-relative-set-2026a}{complement} of $E$ relative to $A$.