every subset of a measurable set is measurable
Solution 1:
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.
Solution 2:
No. This hold only for sets of measure $0$ assuming the measure is complete. The whole space itself is always measurable, so it would mean every set is measurable in that space whis not true for $\mathbb R$.