On continuity of measurable group representations and homomorphisms
Volume 210 / 2012
Studia Mathematica 210 (2012), 197-208 MSC: Primary 22D10; Secondary 43A05, 28A05, 54H11. DOI: 10.4064/sm210-3-1
Let $G$ be a locally compact group, and let $U$ be its unitary representation on a Hilbert space $H$. Endow the space $\mathcal L(H)$ of bounded linear operators on $H$ with the weak operator topology. We prove that if $U$ is a measurable map from $G$ to $\mathcal L(H)$ then it is continuous. This result was known before for separable $H$. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.