What is outer measure?

An outer measure on $X$ is a map $\mu^* \colon \mathcal P(X) \to [0, \infty]$ with $\mu^*(\varnothing) = 0$, monotone, and countably subadditive. A set $A$ is $\mu^*$-measurable (Carathéodory) if it splits every set additively:

($\leq$ always holds by subadditivity; the content is $\geq$).