  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.
