Is there a measurable set $A$ such that $m(A \cap B)= \frac12 m(B)$ for every open set $B$?
Edit: (t.b.) See also A Lebesgue measure question for further answers.
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 $A\cap[0,1]$.