Are all measure zero sets measurable?
Yes. Suppose that ${\frak m}^\ast E = 0$. On one hand, the inequality: $${\frak m}^\ast T \leq {\frak m}^\ast(T \cap E) + {\frak m}^\ast(T \cap E^c)$$always holds. On the other hand: $${\frak m}^\ast(T \cap E) \leq {\frak m}^\ast E = 0 \implies {\frak m}^\ast(T \cap E) = 0, \quad {\frak m}^\ast(T \cap E^c) \leq {\frak m}^\ast T$$gives the other inequality.