By sum of two sets I mean $A+B := \{x+y:x \in A \quad y \in B\}$, and there is a tip in a book of real analysis by Zhou Minqiang which says:

“If $A,B$ are Borel sets in $\mathbb{R}^{n}$, $A+B$ may not be a Borel set.”

I want to know some specific examples.(Maybe $\mathbb{R}^{1}$ ?)

Any comments will be helpful.

  • $\begingroup$ I heard that we can construct a closed set in $\mathbb{R}^{2}$ project to a non-Borel set. Is that true? $\endgroup$ – YOTAL Sep 22 at 8:21
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    $\begingroup$ A closed set in $R^2$ is a countable union of compacts. So its projection is a countable union of compact sets. Therefore it is a Borel set. $\endgroup$ – juan Sep 22 at 11:24
  • $\begingroup$ Yes. You are right. Maybe just a Borel set in $\mathbb{R}^{2}$ @juan $\endgroup$ – YOTAL Sep 22 at 11:53
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    $\begingroup$ Yes, there are Borel sets with projection non Borel. Lebesgue, in one of his paper pretended to proof that the projection of a Borel set is a Borel set. Lusin detected the error. This started the Theory of analytic sets. But the examples, I think, are always difficult. $\endgroup$ – juan Sep 22 at 14:59
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    $\begingroup$ @juan It was not Lusin who detected the error, but Souslin, a student of Lusin. This is why projections of Borel sets are called Souslin sets. Then Lusin and Sierpinski developed the major part of the theory of Souslin sets. $\endgroup$ – Piotr Hajlasz Sep 22 at 18:21

This is a result of Erdos and Stone: https://www.ams.org/journals/proc/1970-025-02/S0002-9939-1970-0260958-1/S0002-9939-1970-0260958-1.pdf

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  • $\begingroup$ Thanks a lot. Is there any easy example? $\endgroup$ – YOTAL Sep 22 at 15:57

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