Question: $E$ is measurable, $m(E)< \infty$, and $f(x)=m[(E+x)\bigcap E]$ for all > $x \in \mathbb{R}$. Prove $\lim_{x \rightarrow \infty} f(x)=0$.
First, since measure is translation invariant, I'm assuming that $(E+x)\bigcap E=E$. But then I had this thought: if $E=\{1,2,3\}$ and $x=1$, then $E+x = \{2,3,4\}$. So the intersection is just a single point. This will have measure zero.
My question is, I'm not sure if this is the right train of thinking. And, if it is, I'm not sure how to make this rigorous.