Dani, S. G. (1985) On invariant finitely additive measures for automorphism groups acting on tori Transactions of the American Mathematical Society, 287 (1). pp. 189-199. ISSN 0002-9947
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Official URL: http://www.ams.org/journals/tran/1985-287-01/S0002...
Related URL: http://dx.doi.org/10.1090/S0002-9947-1985-0766213-0
Abstract
Consider the natural action of a subgroup H of GL(n, Z) on Tn. We relate the H-invariant finitely additive measures on (Tn, L) where L is the class of all Lebesgue measurable sets, to invariant subtori C such that the H-action on either C or Tn/C factors to an action of an amenable group. In particular, we conclude that if H is a nonamenable group acting irreducibly on Tn then the normalised Haar measure is the only H-invariant finitely additive probability measure on (Tn, L) such that μ(R)=0, where R is the (countable) subgroup consisting of all elements of finite order; this answers a question raised by J. Rosenblatt. Along the way we analyse H-invariant finitely additive measures defined for all subsets of Tn and deduce, in particular, that the Haar measure extends to an H-invariant finitely additive measure defined on all sets if and only if H is amenable.
Item Type: | Article |
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Source: | Copyright of this article belongs to American Mathematical Society. |
Keywords: | Invariant Finitely Additive Measures; Invariant Means |
ID Code: | 74505 |
Deposited On: | 16 Dec 2011 09:24 |
Last Modified: | 18 May 2016 18:53 |
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