Naively Haar null sets in Polish groups

Elekes, Márton and Vidnyánszky, Zoltán (2017) Naively Haar null sets in Polish groups. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 446 (1). pp. 193-200. ISSN 0022-247X

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Abstract

Let (G,⋅) be a Polish group. We say that a set X⊂G is Haar null if there exists a universally measurable set U⊃X and a Borel probability measure μ such that for every g,h∈G we have μ(gUh)=0. We call a set X naively Haar null if there exists a Borel probability measure μ such that for every g,h∈G we have μ(gXh)=0. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null. © 2016

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