A Lebesgue measure question 
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Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$? 

Is there a measurable set $E \subset [0,1]$ such that for any $0 < a < b<1$, the Lebesgue measure $$m(E \cap [a,b])= \frac{b-a}{2}?$$
I am stumped and have no idea.

No, there is no such a set.
Since $E$ is measurable (with positive measure) for all $\epsilon > 0$ there exist a sequence of intervals $(I_n)$ in $[0,1]$ such that $E \subset \bigcup I_n$ and
$$
\sum m(I_n) < m(E) + \epsilon.
$$
Hence we obtain that
$$
\frac{3}{4} \sum m(I_n) < m(E).
$$
Thus
$$
\frac{3}{4} \sum m(I_n) < m(E) \leq m(E \cap \bigcup I_n) = m(\bigcup E \cap I_n) \leq \sum m(E \cap I_n),
$$
which implies that there exists positive integer $k$ such that
$$
\frac{3}{4} m(I_k) \leq m(E \cap I_k).
$$
The last inequality contradicts your assumption that for any interval $I$ we have $m(E \cap I) = m(I)/2$.

Whats more, in the exactly same way you can prove general fact:

If $E \subset \mathbb{R}$ is a measurable set with positive measure and $0 \leq c < 1$ than there exists an interval $I$ such that
  $$
m(E \cap I) \geq c \cdot m(I).
$$

