Intersection of sets of positive measure

Consider the integral $$\int m(E\cap (F+y))dy=\iint\mathbb{1}_E(x)\cdot \mathbb{1}_F(x+y)dxdy$$ By the substitution $x\mapsto x$, $y\mapsto z-x$ it becomes $$\iint\mathbb{1}_E(x)\cdot \mathbb{1}_F(z)dxdz=m(E)\cdot m(F)>0$$ This implies that there exists at least one $y$ such that $m(E\cap(F+y))>0$. In fact, we know that there is a set of positive measure of such points.