Let $A,B$ be measurable subsets of $[0,1]$ where $m(A)=m(B)=1/4$. Prove that there exists a $t\in\mathbb{R}$ such that $m(A\cap B_{t})>1/1000$. $B_{t}$ denotes the translation of $B$ by $t$.
Perhaps some kind of set approximation works? as measurable sets can be approximated by "simple" sets (finite unions of intervals)