every subset of a measurable set is measurable Is it true that every subset of a measurable set is measurable? for any measure. So if A is a measurable set then, B as a subset of A must be measurable wrt the same measure.

The answer to that question depends on a choice of the consistent system of axioms in which we prefer to work.
For example, if we consider the question asking  whether every subset of the real axis $R$ is Lebesgue measurable, then we will get the following different  pictures:
Theorem 1 (Vitali(1903)). (ZFC)  There exists a subset of the real axis $R$ which is not measurable in the Lebesgue sense.
Theorem 2 (Mycielski-Swierczkowski(1964)). (ZF+DC+AD) Every subset of the real axis $R$ is measurable in the Lebesgue sense.  
