# Must the Minkowski sum of a Borel set and a *closed* ball be Borel?

Let $A$ be a Borel set in $\mathbb{R}^n$. Must then $A + B(0,1)$ be Borel? Here $B(0,1)$ is the closed ball centered at $0$ of radius $1$.

I know that Erdos and Stone gave an example of a compact set (it is Cantor) and a $G_\delta$-set, whose Minkowski's sum is not Borel. But can we have an example with one of them being a closed ball?

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Also at MathOverflow –  David Moews Feb 9 at 8:37
Could you give the reference for the Erdos-Stone result? –  Nate Eldredge Feb 9 at 13:30
@NateEldredge: P. Erdős and A. H. Stone, On the sum of two Borel sets, Proc. Amer. Math. Soc. 25 (1970), 304-306. –  Martin Feb 9 at 14:34
@AndresCaicedo Thanks for the link. The answer by Tapio Rajala applies to $n\ge 3$ only. What if $n=2$? (The case $n=1$ has an easy affirmative answer). –  user53153 Feb 9 at 19:27