TITLE: Is the Minkowski sum of two Lebesgue measurable set measurable?
QUESTION [7 upvotes]: Let $E,F \subseteq \mathbb R$ be Lebesgue measrable. Is the sum $E+F=\{x+y:x \in E,y \in F \}$ Lebesgue measurable as well?

REPLY [6 votes]: No. Sierpiński showed the existence of measure zero sets $X$, $Y$ whose sum $X + Y$ is non-measurable. See this paper for a detailed discussion of this.