TITLE: Is the sum of 2 Lebesgue measurable sets measurable?
QUESTION [21 upvotes]: Is the sum of two measurable set measurable? I think it is not...

REPLY [6 votes]: Note that the problem is trivial if you talk about subsets of the plane $\mathbb R\times \mathbb R$.  Let $A\subseteq \mathbb R$ be non-measurable, then $A\times \{0\}$ and $\{0\}\times \mathbb R$ both have Lebesgue measure 0 in the plane, but their sum $A\times \mathbb R$ is not measurable.