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 $\mathfrak{fm}(G)$ was introduced, and a positive solution to the Haar Measure Problem for $G$ was given under the assumption that $\mathrm{non}(\mathcal{N})\le \mathfrak{fm}(G)$. We prove here that it is consistent with ZFC that there is a metrizable profinite group $G_{\ast }$ such that $\mathrm{non}(\mathcal{N})>\mathfrak{fm}(G_{\ast })$, thus demonstrating that the strategy of [6] does not suffice for a general solution to the Haar Measure Problem.