
TL;DR
This paper investigates various notions of small sets in abelian Polish groups, focusing on their properties, examples, and the structure of related sigma-ideals, including the non-ideal nature of Haar-countable sets.
Contribution
It provides a comprehensive analysis of Haar-small sets, introduces new examples, and addresses open questions about their algebraic and measure-theoretic properties.
Findings
Haar-countable sets do not form an ideal.
Examples distinguish different smallness notions in all abelian Polish groups.
Answers to questions on null-finite sets are provided.
Abstract
In this paper we are interested in the following notions of smallness: a subset of an abelian Polish group is called Haar-countable/Haar-finite/Haar- if there are a Borel hull and a copy of such that is countable/finite/of cardinality at most , for all . Recently, Banakh et al. have unified the notions of Haar-null and Haar-meager sets by introducing Haar- sets, where is a collection of subsets of . It turns out that if is the -ideal of countable sets, the ideal of finite sets or the collection of sets of cardinality at most , then we get the above notions. Moreover, those notions have been studied independently by Zakrzewski (under a different name -- perfectly -small sets). We study basic properties of the corresponding families of small sets,…
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Taxonomy
TopicsAdvanced Topology and Set Theory
