On continuity of measurable group representations and homomorphisms

TL;DR

This paper proves that measurable unitary representations of locally compact groups are continuous and explores the consistency of the continuity of measurable homomorphisms within set theory, extending known results.

Contribution

It generalizes a theorem on nonmeasurable unions and establishes the continuity of measurable group homomorphisms under certain set-theoretic assumptions.

Findings

Measurable unitary representations are continuous.

Continuity of measurable homomorphisms is consistent with ZFC.

Existence of non-measurable sets related to group actions.

Abstract

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it is continuous. This result was known before for separable H. To prove this, we generalize a known theorem on nonmeasuralbe unions of point finite families of null sets. We prove also that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous. This relies, in turn, on the following theorem: it is consistent with ZFC that for every null set S in a locally compact group there is a set A such that AS is non-measurable.