Automatic continuity of group homomorphisms

Abstract: 

This talk allows to observe some connections of the functional analysis to subtle measure theory and even to the axioms of the set theory. 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. 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.