On invariant finitely additive measures for automorphism groups acting on tori

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
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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