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What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$

I know that neither the Borel $\sigma$-algebra nor the Lebesgue $\sigma$-algebra work, but it is already unclear to me what happens with $\sigma(\mathbf{\Sigma}^1_1(\Bbb R))$, the $\sigma$-algebra generated by the analytic sets (of course the sum of two analytic sets is analytic, but I'm not sure what happens already with $\mathbf{\Pi}^1_1$ sets).

I'm interested in this question for arbitrary perfect Polish groups, but let's start with the (hopefully) easiest case.

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  • $\begingroup$ I take it the trivial upper bound is $\mathbf{\Delta}^2_1$? $\endgroup$
    – Emil Jeřábek
    Commented Nov 19, 2021 at 11:09
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    $\begingroup$ If you start with the open sets, then generate using sigma-algebra operations and sums, do you get at least all the analytic sets? $\endgroup$
    – Gerald Edgar
    Commented Nov 19, 2021 at 13:13
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    $\begingroup$ I might be going off of a wild guess, but I suspect an argument like the one here might be able to get that in appropriate sense such sets are closed under projections. This way perhaps one can show this class is equal to the class of $\sigma$-projective sets. $\endgroup$
    – Wojowu
    Commented Nov 19, 2021 at 13:15
  • $\begingroup$ That seems like a very good guess @Wojowu, let's see if I can work out the details $\endgroup$
    – Alessandro Codenotti
    Commented Nov 21, 2021 at 16:03

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