Dani, S. G. (1985) On invariant finitely additive measures for automorphism groups acting on tori Transactions of the American Mathematical Society, 287 (1). pp. 189199. ISSN 00029947

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Official URL: http://www.ams.org/journals/tran/198528701/S0002...
Related URL: http://dx.doi.org/10.1090/S00029947198507662130
Abstract
Consider the natural action of a subgroup H of GL(n, Z) on T^{n}. We relate the Hinvariant finitely additive measures on (T^{n}, L) where L is the class of all Lebesgue measurable sets, to invariant subtori C such that the Haction on either C or T^{n}/C factors to an action of an amenable group. In particular, we conclude that if H is a nonamenable group acting irreducibly on T^{n} then the normalised Haar measure is the only Hinvariant finitely additive probability measure on (T^{n}, 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 Hinvariant finitely additive measures defined for all subsets of T^{n} and deduce, in particular, that the Haar measure extends to an Hinvariant 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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