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

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 $B\subset X$ 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 $\{f\in C(K,X):\forall x\in X{f}^{-1}(B+x)\in \mathcal{I}\}$ is comeager in the function space $C(K,X)$. 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 hull of Haar-$\mathcal{I}$ sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasǎk on Borel hulls of Haar-null and Haar-meager sets. Also we establish various Steinhaus properties of the families of (generically) Haar-$\mathcal{I}$ sets in Polish groups for various ideals $\mathcal{I}$.

71 pages

References in corpus (2)

• The Steinhaus property and Haar-null sets
• On $\infty$-convex sets in spaces of scatteredly continuous functions

Cited by in corpus (1)

• Error recognition in the Cantor cube