I have this tough problem that I could use some help with. Consider two Lebesgue integrable sets $A,B$ such that $\infty>\mu(A)>0\mu(B)>0$. Prove we can find $x\in \mathbb{R}$ such that $\mu(A\cap(x+B))>0$.
Since we are to prove something exissts, I'm guessing the right way to approach this is by contradiction. So I let $\mu(A\cap(x+B))=0$ for all $x\in\mathbb{R}$. This means $\int_{A\cap(x+B)}d\mu=0$ for all $x$, or that $\chi_A *\chi_{-B}$ has zero integral.
Is this right? Does this lead somewhere? Any help...? Any hints? Thanks in advance...