Nonmeasurable subgroups of compact groups

TL;DR

This paper investigates the existence of non-measurable and non-Borel subgroups within compact groups, providing affirmative results for most classes except certain metric profinite groups, thus advancing understanding of subgroup measurability.

Contribution

It proves that all compact groups contain non-Borel subgroups and identifies conditions under which non-measurable subgroups exist, except in some strongly complete metric profinite groups.

Findings

Every compact group contains a non-Borel subgroup.

Most compact groups have a non-measurable subgroup.

Some metric profinite groups are exceptions with no non-measurable subgroups.

Abstract

In 1985 S.~Saeki and K.~Stromberg published the following question: {\it Does every infinite compact group have a subgroup which is not Haar measurable?} An affirmative answer is given for all compact groups with the exception of some metric profinite groups known as strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup.