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.
Highlights
We survey results about Haar null subsets of Polish groups
In the paper [23], Christensen introduced the notion of Haar null sets, which is equivalent to having Haar measure zero in locally compact groups and defined in every abelian Polish group. (A topological group is Polish if it is separable and completely metrizable, for the definition of Haar nullness see Definition 3.1.1.) Since lots of papers were published which either study some property of Haar null sets or use this notion of smallness to state facts which are true for almost every element of some structure
Haar meager sets were first introduced by Darji in [29] in 2013 as a topological counterpart to the Haar null sets. (Meagerness remains meaningful in non-locally compact groups, but Haar meager sets are a better analogue for Haar null sets.) This original definition only considered the case of abelian groups, but it was straightforwardly generalized in [38] to work in arbitrary Polish groups
Summary
This section is the collection of the miscellaneous notations, definitions, and conventions that are used repeatedly in this paper. (Note that this convention allows us to write the Cantor set as 2ω = {0, 1}ω.) If X is a topological space, . Most of the results in this paper are about certain subsets of Polish groups. (A metric d on G is called two-sided invariant (or invariant) if d(g1hg, g1kg2) = d(h, k) for any g1, g2, h, k ∈ G.) Groups with this property are called TSI groups This class of groups properly contains all Polish, abelian groups, since each metric group G admits a left-invariant metric which, obviously, is invariant when G is abelian. Any invariant metric on a Polish group is automatically complete For proofs of these facts and more results about TSI groups, see, for example, [58, § 8]. If we assume that G is locally compact, our notions will coincide with simpler notions (see Subsection 3.3) and the majority of the results in this paper become significantly easier to prove, so the interesting case is when G is not locally compact
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