Hi,


Let $R$ be equipped with the usual Borel structure. Let $F$ be a Borel subset and $E$ be a closed subset of $R$. Then $F+E=(f+e: f\in F, e \in E \)$ is Borel? If yes, is it true for any locally compact topological group? Thanks in advance.



Borel set plus a closed set = Borel
No. Erdös and Stone showed that the sum of two subsets $E$, $F\subset\mathbb R$ may not be Borel even if one of them is compact and the other is $G\_\delta$ (see [*"On the Sum of Two Borel Sets"*](http://www.jstor.org/pss/2037209), Proc. Am. Math. Soc., Vol. 25, (1970), pp. 304-306). 


Their argument works for every connected locally compact (or abelian) topological group with a complete metric.


