Continuity of measure under translation in $\mathbb{R}^n$

Question

Let $A,B\subseteq\mathbb{R}^n$ be Lebesgue measurable such that at least one of them has finite measure. Then the function $f:\mathbb{R}^n\to\mathbb{R}$ defined by $f(x)=\mu(B\cap(A+x))$ is continuous.

This is just an unanswered question I stumbleuponed here that interests me. Could anyone help answering here?

Answer 1

This can be solved in a similar way as in the linked question.

1. We first reduce to the case where both $A$ and $B$ are Borel sets have finite measure.
2. Then we approximate $A$ and $B$ by finite unions of sets of the form $\prod_{j=1}^nA_j$ and $\prod_{j=1}^nB_j$, where $A_j$ and $B_j$ are Borel sets of finite measure.
3. We thus are reduced to the case $n=1$.