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Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

Volume 564 / 2021

Taras Banakh, Szymon Głąb, Eliza Jabłońska, Jarosław Swaczyna Dissertationes Mathematicae 564 (2021), 1-105 MSC: Primary 54H05, 54H11; Secondary 03E15, 22A10, 28C10. DOI: 10.4064/dm812-2-2021 Published online: 9 August 2021

Abstract

Generalizing Christensen’s notion of a Haar-null set and Darji’s notion of a Haar-meager set, we introduce and study the notion of a Haar-$\mathcal I$ set in a Polish group. Here $\mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $B\subseteq X$ of a Polish group $X$ is called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that $f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically Haar-$\mathcal I$ if the set of witness functions $\{f\in \mathcal C(K,X):\forall x\in X f^{-1}(B+x)\in\mathcal I\}$ is comeager in the function space $\mathcal C(K,X)$. We study (generically) Haar-$\mathcal I$ sets in Polish groups for many concrete and abstract ideals $\mathcal I$, and construct the corresponding distinguishing examples. We prove some results on Borel hulls of Haar-$\mathcal I$ sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasák on Borel hulls of Haar-null and Haar-meager sets. We also establish various Steinhaus properties of the families of (generically) Haar-$\mathcal I$ sets in Polish groups for various ideals $\mathcal I$.

Authors

  • Taras BanakhJan Kochanowski University in Kielce
    Poland
    and
    Ivan Franko National University in Lviv
    Ukraine
    e-mail
  • Szymon GłąbInstitute of Mathematics
    Łódź University of Technology
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail
  • Eliza JabłońskaAGH University of Science and Technology
    Faculty of Applied Mathematics
    Mickiewicza 30
    30-059 Kraków, Poland
    e-mail
  • Jarosław SwaczynaInstitute of Mathematics
    Łódź University of Technology
    Wólczańska 215
    93-005 Łódź, Poland
    and
    Mathematical Institute
    Czech Academy of Sciences
    Žitná 25
    115 67 Praha 1, Czech Republic
    e-mail
    e-mail

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