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arXiv · 1010.0999

On continuity of measurable group representations and homomorphisms

Yulia Kuznetsova
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Abstract

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.

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Yulia Kuznetsova. 2015-09-14. On continuity of measurable group representations and homomorphisms. https://doi.org/10.4064/sm210-3-1

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