Every compact group can have a non-measurable subgroup

TL;DR

This paper proves that it is consistent with ZFC that every compact group contains a non-measurable subgroup, providing a natural construction and confirming it in the Abelian case.

Contribution

It introduces a natural construction for non-measurable subgroups in compact groups and proves its validity in the Abelian case, advancing understanding of measure theory in group structures.

Findings

Every compact group can have a non-measurable subgroup under ZFC consistency.

A natural construction for such subgroups is demonstrated.

The construction is proven to produce non-measurable subgroups in Abelian compact groups.

Abstract

We show that it is consistent with ZFC that every compact group has a non-Haar-measurable subgroup. In addition, we demonstrate a natural construction, and we conjecture that this construction always produces a non-measurable subgroup of a given compact group. We prove that this is so in the Abelian case.