Carathéodory's Theorem (Measure Theory)

Theorem
Let $X$ be a set.

Let $\SS \subseteq \powerset X$ be a semi-ring of subsets of $X$.

Let $\mu: \SS \to \overline \R$ be a pre-measure on $\SS$.

Let $\map \sigma \SS$ be the $\sigma$-algebra generated by $\SS$.

Then $\mu$ extends to a measure $\mu^*$ on $\map \sigma \SS$.