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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Irregular invariant measures related to Haar measure


Author: H. LeRoy Peterson
Journal: Proc. Amer. Math. Soc. 24 (1970), 356-361
MSC: Primary 28.75
DOI: https://doi.org/10.1090/S0002-9939-1970-0249575-7
Erratum: Proc. Amer. Math. Soc. 42 (1974), 645.
MathSciNet review: 0249575
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Abstract: Let G be a locally compact nondiscrete group, and let $ {\nu _1}$ be a Haar measure on an open subgroup of $ G$. It is not hard to show that $ {\nu _1}$ must be the restriction of a Haar measure $ \nu $ on all of $ G$. Here we show that there exists a translation invariant measure $ \mu $ (found by extending $ {\nu _1}$ to the cosets of $ H$ in a natural way) which agrees with $ \nu $ on, for example, $ (\nu )$ $ \sigma $-finite sets, open sets, and subsets of $ H$. Although $ \nu $ can be computed from $ \mu $ in a relatively simple manner, the two measures are not equal in general. In fact, there is an extreme case, namely when $ H$ is not $ \sigma $-compact and has uncountably many cosets, in which $ \mu $ fails very badly to be regular--there are closed sets on which $ \mu $ is not inner regular and (other) closed sets on which $ \mu $ is not outer regular. One condition sufficient for this extreme case to be possible is when $ G$ is Abelian and not $ \sigma $-compact.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0249575-7
Keywords: Locally compact group, Abelian group, translation-invariant measure, regular measure, $ \sigma $-compact space
Article copyright: © Copyright 1970 American Mathematical Society