Let $E$ have finite (Lebesgue) outer measure. Now we need to show that if $E$ is not measurable, then there is an open set $O$ containing $E$ that has finite outer measure and for which $m^*(O-E)>m^*(O)-m^*(E)$. (Here $m^*(E)$ denotes the Lebesgue outer measure of $E$). The following is my attempt.
Suppose $E$ is not measurable. Assume that for all open sets $O$ containing $E$ that has finite outer measure we have $m^*(O-E)\leq m^*(O)-m^*(E)$. Let $\epsilon >0$. Since $m^*(E)=\inf\{m^*(U)|E\subseteq U\ \text{and}\ U\ \text{is open}\}$, $\exists U\ \text{open with}\ E\subseteq U\ \text{such that}\ m^*(U)-m^*(E)<\epsilon$. Then $U$ has finite outer measure. So by assumption it follows that $m^*(U-E)<\epsilon$. Therefore $E$ is measurable; contradiction. Hence the result.
Could someone please tell me if this proof is alright? Thanks.
Best Answer
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.