On a cardinal invariant related to the Haar measure problem

Abstract

In [6], given a metrizable profinite group G, a cardinal invariant of the continuum fm(G) was introduced, and a positive solution to the Haar Measure Problem for G was given under the assumption that non(N ) fm(G). We prove here that it is consistent with ZFC that there is a metrizable profinite group G * such that non(N ) > fm(G * ), thus demonstrating that the strategy of [6] does not suffice for a general solution to the Haar Measure Problem.