A question about measurable sets with positive measure?

Question

Let $E,F$ be measurable sets with positive measure. Does there exist a translate of $F$ such that it intersects $E$ with positive measure? I feel like this should be true, but can’t think of a proof.

Answer 1

Fubini's Theorem yields a simple proof: if the conclusion is false then $\int \mu (E \cap (F+x)) dx=0$ Write$\mu (E \cap (F+x)) =\int I_E (y) I_{F+x} (y) \, d\mu (y)$ and interchange the two integrals. You get $\int \int I_{F+x} (y) \, dx I_E (y) \, dy$. Note that $I_{F+x} (y) \equiv I_{y-F} (x)$ . Also Lebesgue measure of $y-F$ is same as Lebesgue measure of $F$. Finally we get $\mu (E) \mu (F)=0$ which is a contradiction.