Intersection of a Borel set and its translation

Question

My idea is to use regularity somehow and use compact sets where this statement clearly holds. This doesn't seem to be working. Is this the right approach? Any help is appreciated.

Answer 1

Choose $\epsilon>0$. Since $mB<\infty$, there is some $L$ such that $m (B \setminus [-L,L]^n) < \epsilon$. Choose $\|x\|_\infty > 2L$. Then $m(B \cap (B + x)) < 2 \epsilon$.

A little elaboration:

Let $B_0 = B \cap [-L,L]^n, B_1 = B \setminus [-L,L]^n$. Note that if $\|x\|_\infty > 2L$ then $B_0 \cap (B_0 +x) = \emptyset$. Find an upper bound for the measure of $(B_0 \cup B_1) \cap ((B_0+x) \cup (B_1+x))$ by expanding into four sets.