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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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    Yes (and in this case, it would be better to write $\mu^*(A\cap E^c\cap F)$ in order make make this clearer)2012-09-12
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    $A \cap E^c = (A \cap F \cap E^c) \cup (A \cap E^c \cap F^c)$. Is your question about this statement or about the theorem in the title?2012-09-12
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    What I wrote is required to understand the proof of the theorem I wrote in the title. I understand everything now -- thanks for the help.2012-09-12
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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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