A theorem of Piccard’s type in abelian Polish groups. (English) Zbl 1389.28025
The main result of this paper is: Let \((X, +, d)\) be an abelian Polish group. Let \(A\subset X\) be a Borel set which is not Haar meager. Then \(F_N(A):=\bigg\{(x_1,\dots,x_N) \in X^N:A\cap \bigcap\limits_{i=1}^{N} (A+x_1)\) is not Haar meager in \( X\bigg\}\) is a neighbourhood of \(0\) in \(X^N\) for every \(N\in\mathbb{N}\).
Reviewer: Ryszard Pawlak (Łódź)
MSC:
| 28E05 | Nonstandard measure theory |
| 28C10 | Set functions and measures on topological groups or semigroups, Haar measures, invariant measures |
| 54B30 | Categorical methods in general topology |
| 54E52 | Baire category, Baire spaces |
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