Basic problem about measurable sets

Question

Problem

Let $E \subset \mathbb R^n$ with $E$ a measurable set, $E=A \cup B$, where $ |B|=0$. Show that $A$ is measurable.

Here is what I could do (btw, $ |.|_e$ stands for outer measure)

We have $$A=(A \setminus B) \cup (A \cap B)$$$$=(E\setminus B) \cup (A \cap B).$$

Since $E$ and $B$ are measurable, then $E \setminus B$ is measurable, and $A \cap B \subset B$, so $$ |A \cap B|_e\leq |B|_e=0$$

From the last inequality it follows $A \cap B$ is a null set, so it is measurable. Since union of measurable sets is measurable, $A$ is measurable.

Answer 1

As stated, the statement is false. Simply take $E$ to be any nonmeasurable set plus a singleton point, and let $B$ be that singleton point. Then $A$ is most definitely not measurable.

You need the additional condition that $E$ is measurable.