On a Cardinal Invariant Related to the Haar Measure Problem

TL;DR

This paper shows that a previously proposed approach to solving the Haar Measure Problem for certain groups is not universally applicable by demonstrating the consistency of a counterexample within ZFC.

Contribution

It proves the existence of a metrizable profinite group where the key invariant exceeds a certain cardinal, challenging the sufficiency of earlier strategies for the Haar Measure Problem.

Findings

Demonstrates the consistency of $ ext{non}( ext{N}) > rak{fm}(G_*)$ within ZFC.

Shows the previous approach does not solve the Haar Measure Problem in all cases.

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