Irregular invariant measures related to Haar measure
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- by H. LeRoy Peterson
- Proc. Amer. Math. Soc. 24 (1970), 356-361
- DOI: https://doi.org/10.1090/S0002-9939-1970-0249575-7
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Erratum: Proc. Amer. Math. Soc. 42 (1974), 645.
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.References
- S. K. Berberian, Counterexamples in Haar measure, Amer. Math. Monthly 73 (1966), no. 4, 135–140. MR 195984, DOI 10.2307/2313767
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387
- H. LeRoy Peterson, Regular and irregular measures on groups and dyadic spaces, Pacific J. Math. 28 (1969), 173–182. MR 240236
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 24 (1970), 356-361
- MSC: Primary 28.75
- DOI: https://doi.org/10.1090/S0002-9939-1970-0249575-7
- MathSciNet review: 0249575