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-I set in a Polish group. Here I is an ideal of subsets of some compact metrizable space K. A Borel subset B ⊆ X of a Polish group X is called Haar-I if there exists a continuous map f : K → X such that f −1 (B + x) ∈ I for all x ∈ X. Moreover, B is generically Haar-I if the set of witness functions {f ∈ C(K, X) : ∀x ∈ X f −1 (B + x) ∈ I} is comeager in the function space C(K, X). We study (generically) Haar-I sets in Polish groups for many concrete and abstract ideals I, and construct the corresponding distinguishing examples. We prove some results on Borel hulls of Haar-I sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasǎk on Borel hulls of Haar-null and Haar-meager sets. We also establish various Steinhaus properties of the families of (generically) Haar-I sets in Polish groups for various ideals I.Acknowledgements. We would like to express our sincere thanks to Sławomir Solecki for clarifying the situation with the unpublished result of Hjorth [34] (on the Σ 1 1 -hardness of the family CND of closed non-dominating sets) and sending us a copy of the handwritten notes of Hjorth.Special thanks are due to the anonymous referee for careful reading of the manuscript and many valuable remarks and suggestions.

The Set-Cover game and nonmeasurable unions

“…The family A \ I is defined to have the weak Steinhaus property if there exists n ∈ N such that for any set A ∈ A \ I in X the set (AA −1 ) n is a neighborhood of the identity in X. More information of the (weak) Steinhaus property of ideals can be found in [4].…”

Products of K-Analytic Sets in Locally Compact Groups and Kuczma–Ger Classes

“…Theorem 1 follows from Lemmas 6 and 7, proved in this section. In the proof of Lemma 6, we shall use the following lemma, whose proof goes along the lines of the proof of Theorem 3.2 in [12]. Lemma 5.…”

“…A subset A of a topological group X is called Haar-null if there exists a Borel set B ⊇ A in X and a probability Radon measure µ on X such that µ(xBy) = 0 for any x, y ∈ X. Haar-null sets were introduced by Christensen [11] who proved that a subset of a locally compact group is Haar-null if and only if its Haar measure is zero. For more information on Haar-null sets and their generalizations, see [12,13]. By ( [12], Example 8.1), the Polish group Z ω contains a non-open Borel subgroup, which cannot be covered by countably many closed Haar-null sets in Z ω .…”

“…For more information on Haar-null sets and their generalizations, see [12,13]. By ( [12], Example 8.1), the Polish group Z ω contains a non-open Borel subgroup, which cannot be covered by countably many closed Haar-null sets in Z ω . By [14], the countable product of lines contains a meager Borel linear subspace L ⊆ R ω that cannot be covered by countably many Haar-null sets in R ω .…”

Haar-smallest sets

“…Recently, the notions of Haar-null sets and Haar-meager sets were unified by Banakh et al in [3] by introducing the concept of Haar-small sets. For a semi-ideal I on the Cantor cube 2 ω , we say that a subset A of an abelian Polish group X is Haar-I (A ∈ HI) if there are a Borel hull B ⊇ A and a continuous map f : 2 ω → X with f −1 [B + x] ∈ I for all x ∈ X.…”