Extensions of Haar measure to relatively large nonmeasurable subgroups

Author:
H. Leroy Peterson

Journal:
Trans. Amer. Math. Soc. **228** (1977), 359-370

MSC:
Primary 43A05; Secondary 22D05, 28A70

DOI:
https://doi.org/10.1090/S0002-9947-1977-0435728-2

MathSciNet review:
0435728

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Abstract: Let *G* be a locally compact group, with a left Haar measure on *G*. A subgroup is *large* if it has finite index; a *relatively large* subgroup of *G* is a large subgroup of an open subgroup. In §1 we have an existence theorem for relatively large nonopen subgroups, and we observe that such subgroups are not -measurable. This motivates the development, in §2, of a left-invariant countably additive extension of , which includes in its domain all unions of left translates of a given relatively large subgroup *K*. For an arbitrarily chosen family of relatively large subgroups of *G*, we define (in §3) a finitely additive measure such that, for any is an extension of the corresponding defined in §2. An example shows that need not be countably additive. Finally, in §4, we observe some aspects of the relationship between -measurable and -measurable functions, in the context of existing literature on extensions of Haar measure. In particular, we generalize the well-known proposition that -measurable characters are continuous.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0435728-2

Keywords:
Subgroups of finite index,
locally compact group,
extension of Haar measure,
sets of full outer measure,
characters on an Abelian group

Article copyright:
© Copyright 1977
American Mathematical Society