Haar measure for non-Hausdorff locally compact groups

TL;DR

This paper explores extending Haar measure to non-Hausdorff locally compact groups by proposing two approaches, one of which generalizes classical theorems to broader group classes.

Contribution

It introduces two methods for defining Haar measure in non-Hausdorff groups, including a generalization of existence and uniqueness theorems.

Findings

Counterexample to Haar measure existence with compact sets measurable

Generalization of Haar measure existence and uniqueness

Two distinct approaches to measure extension

Abstract

The paper describes two possible ways of extending the definition of Haar measure to non-Hausdorff locally compact groups. The first one forces compact sets to be measurable: with this construction, a counterexample to the existence of the Haar measure is provided. The second one makes use of closed compact sets instead of compact sets in the definition of Radon measure: this way, the classical theorems of existence and uniqueness of the Haar measure can be generalised to locally compact groups, not necessarily Hausdorff.