The Haar Measure Problem

TL;DR

This paper proves that under a weak form of the Continuum Hypothesis, every infinite metrizable profinite group has a Haar-nonmeasurable subgroup, addressing a longstanding open problem in group theory.

Contribution

It demonstrates the existence of Haar-nonmeasurable subgroups in all infinite metrizable profinite groups assuming a weak continuum hypothesis.

Findings

Existence of Haar-nonmeasurable subgroups under the hypothesis

Dual Baire category analogue established

Addresses the Haar Measure Problem in specific groups

Abstract

An old problem asks whether every compact group has a Haar-nonmeasurable subgroup. A series of earlier results reduce the problem to infinite metrizable profinite groups. We provide a positive answer, assuming a weak, potentially provable, consequence of the Continuum Hypothesis. We also establish the dual, Baire category analogue of this result.