Naively Haar null sets in Polish groups

Let (G,.) be a Polish group. We say that a set X subset of G is Haar null if there exists a universally measurable set U superset of X and a Borel probability measure mu such that for every g,h epsilon G we have mu,(gUh) = 0. We call a set X naively Haar null if there exists a Borel probability measure A such that for every g, h epsilon G we have mu(gXh) = 0. Generalizing a result of Elekes and Steprans, 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. (C) 2016 Published by Elsevier Inc.