Outer Measure of Limit of Increasing Sequence of Sets

Theorem
Let $\mu^*$ be an outer measure on a set $X$.

Let $\left\langle{S_n}\right\rangle$ be an increasing sequence of $\mu^*$-measurable sets, and let $S_n \uparrow S$ (as $n \to \infty$).

Then for any subset $A \subseteq X$, $\displaystyle \mu^* \left({A \cap S}\right) = \lim_{n \to \infty} \mu^* \left({A \cap S_n}\right)$.