Determining if a set is measurable by upper and lower sets

Question

I have the following question regarding Lebesgue measure: If $A,B$ are measurable sets and I have $m(A\setminus E)=0$ and $m(E\setminus B)=0$, is it enough to determine that $E$ is measurable? We do not know whether $E\subset A$ or $B\subset E$ or any information of that sort, but we do know that $A\cap E\not = \emptyset$ $B\cap E\not = \emptyset$. Thanks for all the answers.

Answer 1

For a counterexample, let $A=\{2\}$, $B=[0,1]$, and let $E$ be the union of $\{2\}$ with a nonmeasurable subset of $[0,1]$. Then

• $A\setminus E$ is empty
• $E\setminus B = \{2\}$
• $A\cap E$ and $B\cap E$ are nonempty