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Transactions of the American Mathematical Society

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



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
MathSciNet review: 766213
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Abstract: Consider the natural action of a subgroup $ H$ of $ {\text{GL}}(n,{\mathbf{Z}})$ on $ {{\mathbf{T}}^n}$. We relate the $ H$-invariant finitely additive measures on $ ({{\mathbf{T}}^n},\mathcal{L})$ where $ \mathcal{L}$ is the class of all Lebesgue measurable sets, to invariant subtori $ C$ such that the $ H$-action on either $ C$ or $ {{\mathbf{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 $ {{\mathbf{T}}^n}$ then the normalised Haar measure is the only $ H$-invariant finitely additive probability measure on $ ({{\mathbf{T}}^n},\mathcal{L})$ such that $ \mu (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 $ {{\mathbf{T}}^n}$ 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.

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Keywords: Invariant finitely additive measures, invariant means
Article copyright: © Copyright 1985 American Mathematical Society