I am trying to prove the following: For a Borel set $A$ with finite positive Lebesgue measure prove that $\exists t$ s.t $\forall x \in (-t,t)$, $A\bigcap (A-x)\neq \emptyset$
I'm trying to do this using the indicator function on A: $\chi _{A}$ and the fact that $lim_{x\rightarrow 0}\int |\chi _{A}(t-x)-\chi _{A}(t)|dt=0$ but am getting confused over how the limit going to 0 can help achieve the desired solution.