Measures invariant under a linear group
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- by Larry Baggett PDF
- Proc. Amer. Math. Soc. 94 (1985), 179-186 Request permission
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
- D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74 (Russian). MR 0209390
- Dennis Sullivan, For $n>3$ there is only one finitely additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable subsets, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 121–123. MR 590825, DOI 10.1090/S0273-0979-1981-14880-1
Additional Information
- © Copyright 1985 American Mathematical Society
- 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