Carathéodory measurable set

A Carathéodory measurable set (with respect to an outer measure $\mu^*$ on $X$ ) is a subset $E\subseteq X$ such that for every subset $S\subseteq X$ ,

\mu^*(S)=\mu^*(S\cap E)+\mu^*(S\setminus E).

This condition says $E$ “splits” every set $S$ without loss of outer measure, using intersection and set difference. In the Carathéodory construction, the collection of Carathéodory measurable sets forms a sigma-algebra, and restricting $\mu^*$ to it gives a measure.

Examples

If $\mu^*$ is induced from a measure $\mu$ on a sigma-algebra $\Sigma$ (via the usual infimum over measurable supersets), then every $A\in\Sigma$ is Carathéodory measurable.
For Lebesgue outer measure on $\mathbb R$ , every Borel set is Carathéodory measurable (and in fact many more sets are as well).