On invariant finitely additive measures for automorphism groups acting on tori

Author:
S. G. Dani

Journal:
Trans. Amer. Math. Soc. **287** (1985), 189-199

MSC:
Primary 28D15; Secondary 43A07

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766213-0

MathSciNet review:
766213

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider the natural action of a subgroup of on . We relate the -invariant finitely additive measures on where is the class of all Lebesgue measurable sets, to invariant subtori such that the -action on either or factors to an action of an amenable group. In particular, we conclude that if is a nonamenable group acting irreducibly on then the normalised Haar measure is the only -invariant finitely additive probability measure on such that , where is the (countable) subgroup consisting of all elements of finite order; this answers a question raised by J. Rosenblatt.

Along the way we analyse -invariant finitely additive measures defined for all subsets of and deduce, in particular, that the Haar measure extends to an -invariant finitely additive measure defined on all sets if and only if is amenable.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766213-0

Keywords:
Invariant finitely additive measures,
invariant means

Article copyright:
© Copyright 1985
American Mathematical Society