Automatic continuity of measurable homomorphisms on Cech-complete topological groups

TL;DR

This paper establishes that for Cech-complete topological groups, any Borel, universally, or Haar-measurable homomorphism is automatically continuous, extending classical results in topological group theory.

Contribution

It proves the equivalence of various measurability conditions and continuity for homomorphisms on Cech-complete groups, resolving an open problem and generalizing previous theorems.

Findings

Measurable homomorphisms are continuous in Cech-complete groups.

Extends classical results to broader classes of topological groups.

Answers a problem posed by Kuznetsova.

Abstract

We prove that a homomorphism $h:X\to Y$ from a (locally compact) Cech-complete topological group $X$ to a topological group $Y$ is continuous if and only if $h$ is Borel-measurable if and only if $h$ is universally measurable (if and only if $h$ is Haar-measurable). This answers a problem of Kuznetsova and extends a result of Kleppner on the continuity of Haar-measurable homomorphisms between locally compact groups and a result of Rosendal on the continuity of universally measurable homomorphisms between Polish groups.