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Cardinal invariants of Haar null and Haar meager sets

Part of: Noncompact transformation groups Classical measure theory Set theory

Published online by Cambridge University Press:  27 October 2020

Márton Elekes  and
Márk Poór
Márton Elekes
Affiliation:
Alfréd Rényi Institute of Mathematics, PO Box 127, 1364, Budapest, Hungary and Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, 1117, Budapest, Hungary ([email protected]; http://www.renyi.hu/~emarci)
Márk Poór
Affiliation:
Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, 1117, Budapest, Hungary ([email protected])
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Abstract

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A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) 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.

Keywords

ChristensenHaar nullHaar meagercardinal invariantscardinal characteristicsaddcovnoncofCichoń diagram

MSC classification

Primary: 03E17: Cardinal characteristics of the continuum
Secondary: 22F99: None of the above, but in this section 03E15: Descriptive set theory 28A99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1568 - 1594
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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