Naively Haar null sets in Polish groups

TL;DR

This paper introduces the concept of naively Haar null sets in Polish groups and proves that such sets can exist without being Haar null, expanding understanding of measure-theoretic properties in these groups.

Contribution

It generalizes previous results by showing the existence of naively Haar null sets that are not Haar null in all abelian Polish groups.

Findings

Existence of naively Haar null sets that are not Haar null in abelian Polish groups

Generalization of a result by Elekes and Stepr ext=ans

Answers part of Problem FC from Fremlin's list

Abstract

Let $(G,\cdot)$ be a Polish group. We say that a set $X \subset G$ is Haar null if there exists a universally measurable set $U \supset X$ and a Borel probability measure $\mu$ such that for every $g, h \in G$ we have $\mu(gUh)=0$. We call a set $X$ naively Haar null if there exists a Borel probability measure $\mu$ such that for every $g, h \in G$ we have $\mu(gXh)=0$. Generalizing a result of Elekes and Stepr\=ans, 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.