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.

Certainly not. The set $[0,1]$ is Lebesgue measurable, but (if we assume the Axiom of Choice) it has non-measurable subsets. 
If $A$ is a set of Lebesgue measure $0$, then any subset of $A$ is Lebesgue measurable.
