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Let $\mu^* : \mathcal{P}(X) \rightarrow [0, \infty]$ be an outer measure, and let $M$ denote the set of $\mu^*$-measurable sets.

Let $A \subseteq X$ and let $E,F \in M$.

Why is the following statement true?

$\mu^*(A \cap E^c) = \mu^*(A \cap F \cap E^c) + \mu^*(A \cap E^c \cap F^c)$

EDIT: Is it because $F \in M$, with $(A \cap E^c)$ serving as our "test set"?

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    You can answer your own question :-) (to remove this from the list of unanswered questions).2012-09-13

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Yes, it is a special case of the property defining measurable sets, applied to $F$ against $A\cap E^c$.