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"*][1], Proc. Am. Math. Soc., Vol. 25, (1970), pp. 304-306).


  [1]: http://www.jstor.org/pss/2037209