On continuity of measurable group representations and homomorphisms
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.
Keywords
Cite
@article{arxiv.1010.0999,
title = {On continuity of measurable group representations and homomorphisms},
author = {Yulia Kuznetsova},
journal= {arXiv preprint arXiv:1010.0999},
year = {2021}
}
Comments
The previous version was not final, I update it once noticed