Outer Measure of Limit of Increasing Sequence of Sets

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

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

Then for any subset $A \subseteq X$:
 * $\ds \map {\mu^*} {A \cap S} = \lim_{n \mathop \to \infty} \map {\mu^*} {A \cap S_n}$