Originally Posted by

**paulread** If for some $\displaystyle i:\,m^*E_i = \infty$ then it is trivial so we consider only sets of finite outer measure. Note that

$\displaystyle m^*\left(\displaystyle\bigcup_{k=1}^{\infty}(A \cap E_k)\right) \ge m^*\left(\displaystyle\bigcup_{k=1}^n(A \cap E_k\right)=\displaystyle\sum_{k=1}^n m^*(A \cap E_k)$

$\displaystyle \Longrightarrow \displaystyle\lim_{n\to\infty} m^*\left(\displaystyle\bigcup_{k=1}^{\infty}(A \cap E_k)\right) \ge \displaystyle\lim_{n\to\infty} \displaystyle\sum_{k=1}^n m^*(A \cap E_k)$

$\displaystyle \Longrightarrow m^*\left(\displaystyle\bigcup_{k=1}^{\infty}(A \cap E_k)\right) \ge \displaystyle\sum_{k=1}^{\infty}m^*(A \cap E_k) \ge m^*\left(\displaystyle\bigcup_{k=1}^{\infty}(A \cap E_k)\right) $

where the last inequality on the right comes from the proposition that states outer measure is countably subadditive.

So does this seem reasonable?