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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Measures invariant under a linear group


Author: Larry Baggett
Journal: Proc. Amer. Math. Soc. 94 (1985), 179-186
MSC: Primary 28D15; Secondary 22D40
DOI: https://doi.org/10.1090/S0002-9939-1985-0781078-4
MathSciNet review: 781078
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Abstract: This paper deals with the question of when there exists on a Euclidean space a nontrivial probability measure $ \mu $ which is invariant under a group $ \Gamma $ of integer matrices. Necessary conditions on $ \Gamma $ and the dimension are discussed. It is shown that nontrivial examples do exist, but only in dimensions $ \geqslant 6$. In fact, the only explicit example given is in dimension 10.


References [Enhancements On Off] (What's this?)

  • [1] D. Kazhdan, On a connection of the dual space of the group and the structure of its closed subgroups, Functional Anal. Appl. 1 (1967), 71-74. MR 0209390 (35:288)
  • [2] D. Sullivan, For $ n > 3$ there is only one finitely additiv erotationally invariant measure on the $ n$-sphere defined on all Lebesgue measurable sets, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 121-123. MR 590825 (82b:28035)

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DOI: https://doi.org/10.1090/S0002-9939-1985-0781078-4
Article copyright: © Copyright 1985 American Mathematical Society

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