Haar meager sets revisited

TL;DR

This paper explores Haar meager sets in Polish groups, comparing definitions, extending results to nonabelian groups, and providing examples and decompositions that distinguish Haar meager sets from Haar null sets.

Contribution

It introduces alternative definitions, extends properties to nonabelian groups, and provides new decomposition theorems distinguishing Haar meager and Haar null sets.

Findings

Haar meager sets are meager and form a σ-ideal in general Polish groups.

Examples show differences between Haar meager and Haar null sets.

Decomposition of certain groups into Haar meager and Haar null sets.

Abstract

In the present article we investigate Darji's notion of Haar meager sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar meager sets in nonabelian Polish groups and show that many results, including the facts that Haar meager sets are meager and form a $\sigma$-ideal, are valid in the more general setting as well. The article provides various examples distinguishing Haar meager sets from Haar null sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^\omega$, $\mathbb R^\omega$ or any Banach space can be decomposed into a Haar meager set and a Haar null set. We also establish the stability of non-Haar meagerness under Cartesian product.