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", Proc. Am. Math. Soc., Vol. 25, (1970), pp. 304-306).
Andrey Rekalo
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