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

In many proofs of theorems I find this statement “If exist B, Borel measurable set, which contains a set E, and |BE|=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.