Sum of two closed sets is measurable

I need to prove that if $A$ and $B$ are two closed sets in $\mathbb R^n$, then $A + B$ is an $F_\sigma$.

I started by writing out $A$ and $B$ each as an $F_\sigma$ by intersecting each with balls of center $0$ radius $n$, and taking the union, I don't know if that's a good idea.

Any hint is appreciated. Thanks.

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Your title says that you want to show $A+B$ is measurable. Your post says that you want to show that $A+B$ is $F_{\sigma}$. Of course, the latter implies the former, but which one is it you are actually trying to prove? –  Arturo Magidin Oct 18 '11 at 4:04
+1 Since I'm also curious to know how this (the sum is an F-sigma) can be done. I tried taking the union $A + B = \bigcup_{a\in A}\{a\} + B$, but this is an uncountable union unfortunately. –  ae0709 Oct 18 '11 at 4:07
I think that people here kindly gives you answers to your posts. Have you considered expressing your gratefulness, by accepting the answers? –  leo Oct 18 '11 at 8:31
yeah, the uncountable union does not work,.. we have to go to an argument with compact sets –  alice Oct 18 '11 at 13:08

If $A$ is closed and $B$ is compact, show that $A+B$ is closed. (Hint: Use sequential compactness.) In general, write $B$ as a countable union of compact sets. If you need more detail, please let me know and I will gladly provide it.

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thank you, very helpful! –  alice Oct 18 '11 at 13:07
@alice No problem, I'm glad I helped! –  Nick Strehlke Oct 18 '11 at 19:13