Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$?
Solution 1:
Hint: Lebesgue density theorem.
Alternatively, approximate $A\cap[0,1]$ with a finite union of intervals.
On second thought, those hints are overly complicated. You can use the definition of Lebesgue measure to find an open set $B$ containing $A\cap[0,1]$ with measure close to that of $A\cap[0,1]$.