$M \subseteq \mathbb R$ exist, so that $\lambda^{1}(M \cap I)= \frac{1}{2}\lambda^{1}(I)$ for any bounded Interval $I\subseteq \mathbb R$.
Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$?