SCHWEITZER 1983 Problem 9

Prove that if $E \subset \mathbb{R}$ is a bounded set of positive Lebesgue measure, then for every $u < 1/2$ , a point $x=x(u)$ can be found so that $$|(x-h,x+h) \cap E| \geq uh$$ and $$|(x-h,x+h) \cap (\mathbb{R} \setminus E)| \geq uh$$ for all sufficiently small positive values of $h$ .

K. I. Koljada