Seminarium KMMF "Teoria Dwoistości"

It is known that there exist discontinuous weakly measurable unitary representationsof locally compact groups. We show that if one uses a bit stronger variant ofmeasurability, every measurable unitary representation turns out to be continuous.Another close result depends on the axioms of set theory: under Continuum hypothesis(or certain weaker assumptions) every measurable homomorphism from a locally compactgroup into any topological group is continuous. This relies, in turn, on thefollowing theorem (proved under the same assumptions): for every zero measure set Sin a locally compact group there is a set A such that AS is non-measurable.