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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 0435728
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Abstract: Let G be a locally compact group, with $ \lambda $ 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 $ \lambda $-measurable. This motivates the development, in §2, of a left-invariant countably additive extension $ {\lambda ^ + }$ of $ \lambda $, which includes in its domain all unions of left translates of a given relatively large subgroup K. For an arbitrarily chosen family $ {\mathcal{K}_I}$ of relatively large subgroups of G, we define (in §3) a finitely additive measure $ \lambda _I^ + $ such that, for any $ K \in {\mathcal{K}_I},\lambda _I^ + $ is an extension of the corresponding $ {\lambda ^ + }$ defined in §2. An example shows that $ \lambda _I^ + $ need not be countably additive. Finally, in §4, we observe some aspects of the relationship between $ {\lambda ^ + }$-measurable and $ \lambda $-measurable functions, in the context of existing literature on extensions of Haar measure. In particular, we generalize the well-known proposition that $ \lambda $-measurable characters are continuous.

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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

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