Let $\mu^{\ast}:2^X\rightarrow [0,\infty]$ be an outer measure. Let $A\subseteq X$, $E_k$ be a disjoint countable collection of measurable sets and $E=\bigcup E_k$. Show that

$$\mu^\ast(A\cap E) = \sum\mu^{\ast}(A\cap E_k).$$

I approached this problem by taking the measure of $A$ intersection $E$ to get zero. Since $A$ intersection $E$ is a subset of $E$ and the measure of the complement of $A$ intersection $E$ equal to the measure of $E$. But my problem is how to link them up. Can someone offer a suggestion?

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    $\begingroup$ Your MathJax technique could be greatly improved by just a small amount of work. Take a look at my edits to your question. ${}\qquad{}$ $\endgroup$ – Michael Hardy Feb 2 '15 at 19:00
  • $\begingroup$ Thanks @Michael Hardy. That certainly helps. $\endgroup$ – user212436 Feb 2 '15 at 23:22

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