This exercise appeared in my real analysis test last year, and is still puzzling me since then. Ironically, even the professor doubts if the b part is actually truth (still...)
Let $A \subset \mathbb{R}$
a) If $\displaystyle0
b) If $ \displaystyle m^*(A \cap I) \leq \frac{m^*(I)}{2}$ for every open interval, then $m^*(A) = 0$