Show that if $E$ is not measurable, then there is an open set $O$ containing $E$ that has finite outer measure and for which $m^*(O-E)>m^*(O)-m^*(E)$
Solution 1:
Yes, your proof is correct. Perhaps adding a statement noting that $E$ is measurable if and only if $\forall\ \epsilon > 0\ \exists\ U \supseteq E$ such that $m^*(U \setminus E) < \epsilon$ would improve the answer.