Carathéodory's extension theorem

Remarks

For existence, define $\mu^*(B) = \inf \set{ \sum_n \mu_0(A_n) : A_n \in \mathcal{S}, \ B \subseteq \bigcup_n A_n }$. This is an outer measure, and every set of $\mathcal{S}$ satisfies the Carathéodory splitting condition. Countable additivity of $\mu_0$ gives $\mu^* = \mu_0$ on $\mathcal{S}$. So $\mu^*$ restricted to the Carathéodory-measurable sets is the required extension. Uniqueness follows from Dynkin's $\pi$-$\lambda$ theorem, since $\mathcal{S}$ is closed under finite intersections. Example: the half-open intervals $(a, b]$ form a semiring on $\R$, and $\mu_0((a, b]) = b - a$ is a premeasure; the theorem gives Lebesgue measure on the Borel sets, and it is unique. Uniqueness can fail without the finiteness hypothesis in clause 2. On $\Q$, take the semiring of sets $(a, b] \cap \Q$ and the premeasure that is $\infty$ on nonempty sets and $0$ on $\varnothing$. It extends both to counting measure and to $2$ times counting measure on the power set of $\Q$.

Going deeper

For a student

This is the main tool to construct measures: Lebesgue measure, Lebesgue–Stieltjes measures, product measures and measures on sequence spaces all come from it. The only work left in each case is to check that the given set function is countably additive on the semiring. A version for an algebra of sets, with the extension given by an explicit outer measure, is the Hahn–Kolmogorov theorem. The proof goes through an outer measure and Carathéodory's criterion.