Notes on Haar Null Sets

This informal set of notes contains some of the new results which will appear in the author’s 2013 dissertation. We show that every infinite product of locally compact non-compact groups decomposes into the disjoint union of a Haar null set and a meager set, which gives a partial positive answer to a question of Darji. We also show that the compact sets in each such product group are always Haar null, and that the same property holds for the permutation group S∞; these results may be compared with a similar result of Dougherty for groups which admit a two-sided invariant metric. We also provide a number of examples of groups where the family of openly Haar null sets is a proper subclass of the family of Haar null sets, which resolves an uncertainty of Solecki. Other related results are peppered throughout. 1. Basic Definitions. Definition 1. A topological group is a group endowed with a topology for which both the group multiplication map · : G × G → G and the group inversion map −1 : G → G are continuous. A group is Polish if its topology is Polish, i.e. completely metrizable and separable. A group is called locally compact if it admits a base of topology at the identity consisting of compact sets. Every locally compact group G admits a left Haar measure μ, i.e. a regular Borel measure which satisfies μ(gA) = μ(A) for every g ∈ G and every measurable set A ⊆ G, and the left Haar measure is unique up to multiplication by a scalar. Each locally compact group also admits an analogously defined right Haar measure which is also unique up to scalar multiplication. The left and right Haar measures on G need not in general be scalar multiples of one another, but they are always absolutely continuous with respect to one another, i.e. they define the same family of measure zero sets. Let G be a locally compact group and let μ be a left Haar measure on G. For every element g ∈ G, we may define a new measure μg on G by the rule μg(A) = μ(Ag) for every measurable set A. It is easy to check that μg is another left Haar measure, and so there exists some positive real scalar ∆(g) for which μg = ∆(g)μ. This function ∆ : G → R is called the modular function of G and it defines a continuous homomorphism from G into the multiplicative group of positive reals. A group is called unimodular if its modular function is identically 1. A group is unimodular if and only if its left and right Haar measures are scalar multiples of one another. The following definition may be attributed originally to J. P. R. Christensen who first coined the term for abelian Polish groups in [1], but in truth this modern definition has evolved into its current state via the contributions of Topsøe/Hoffman-Jørgenson [10], Hunt/Sauer/Yorke [4], and Mycielski [6]. Definition 2 (Christensen 1972 [1]). Let G be a complete metric group. We say that a universally measurable set A ⊆ G is Haar null if there exists a Borel probability measure μ on G such that μ(gAh) = 0 for all g, h ∈ G. date: October 24, 2012. 1