Haar null and Haar meager sets: a survey and new results

TL;DR

This paper surveys the theory of Haar null and Haar meager sets in Polish groups, consolidating fundamental properties, techniques, and introducing new ideas and generalizations in this area of descriptive set theory.

Contribution

It provides a comprehensive overview of Haar null and Haar meager sets, including new results and generalizations beyond previous work.

Findings

Haar null and Haar meager sets share many properties.

Introduction of Haar meager sets as an analogue to Haar null sets.

Generalization of some results to broader classes of Polish groups.

Abstract

We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.