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- W4256644402 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 regularâthere 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." @default.
- W4256644402 created "2022-05-12" @default.
- W4256644402 creator A5039095948 @default.
- W4256644402 date "1970-02-01" @default.
- W4256644402 modified "2023-09-27" @default.
- W4256644402 title "Irregular Invariant Measures Related to Haar Measure" @default.
- W4256644402 doi "https://doi.org/10.2307/2036361" @default.
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