On continuity of measurable group representations and homomorphisms

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 bounded linear operators on H with the 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. 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.