Cardinal invariants of Haar null and Haar meager sets

Márton Elekes,Márk Poór

doi:10.1017/prm.2020.73

Abstract

A subsetXof a Polish groupGisHaar nullif there exists a Borel probability measure μ and a Borel setBcontainingXsuch that μ(gBh) = 0 for everyg,h∈G. A setXisHaar meagerif there exists a compact metric spaceK, a continuous functionf:K→Gand a Borel setBcontainingXsuch thatf−1(gBh) is meager inKfor everyg,h∈G. We calculate (inZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case$G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

Highlights

Small sets play a fundamental role in many branches of mathematics
Christensen proved that these sets form a proper σ-ideal which coincides with the family of sets of Haar measure zero in the locally compact case
Note that if G is locally compact the Haar null sets agree with the sets of Haar measure zero and Haar meager sets agree with the meager sets [9]

Summary

Introduction

Small sets play a fundamental role in many branches of mathematics. Perhaps the most important example is the family of nullsets of a natural invariant measure. Christensen proved that these sets form a proper σ-ideal which coincides with the family of sets of Haar measure zero in the locally compact case. This notion turned out to be very useful in various branches of mathematics, see e.g. the survey paper [9]. To the Haar null case, Darji proved that these sets form a proper σ-ideal which coincides with the family of meager sets in the locally compact case. A subset X of a Polish group G is generalized Haar null if there exists a (completed) Borel probability measure μ and a universally measurable set B containing X such that μ(gBh) = 0 for every g, h ∈ G. This latter holds e.g. for Zω and for separable infinite dimensional Banach spaces (Zω admits a continuous homomorphism onto (Z/2Z)ω, and Banach spaces admit continuous homomorphisms onto their finite dimensional subspaces)

Covering number and uniformity

Cofinality