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Automatic Continuity of Measurable Group Homomorphisms

Jonathan W. Lewin
Proceedings of the American Mathematical Society
Vol. 87, No. 1 (Jan. - Apr., 1983), pp. 78-82
DOI: 10.2307/2044356
Stable URL: http://www.jstor.org/stable/2044356
Page Count: 5
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Automatic Continuity of Measurable Group Homomorphisms
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Abstract

It is well known that a measurable homomorphism from a locally compact group $G$ to a topological group $Y$ must be continuous if $Y$ is either separable or $\sigma$-compact. In this work it is shown that the above requirement on $Y$ can be somewhat relaxed and it is shown inter alia that a measurable homomorphism from a locally compact group to a locally compact abelian group will always be continuous. In addition, it is shown that if $H$ is a nonopen subgroup of a locally compact group, then under a variety of circumstances, some union of cosets of $H$ must fail to be measurable.

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