If a set E is contained in a Borel measurable set B, and the Lebesgue measure of B\E is 0, then is E a Lebesgue measurable set?

Question

In many proofs of theorems I find this statement "If exist $B$, Borel measurable set, which contains a set $E$, and $|B \setminus E|=0$, then $E$ is Lebesgue measurable because of the completeness of the Lebesgue measure". I had to report the question because I asked the wrong question.

Answer 1

Set $K=B\setminus E$. Since $K$ has mesure $0$ it's Lebesgue measurable. Since $E=B\cap K^c$ then $E$ is Lebesgue measurable.