Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

A note on invariant finitely additive measures


Author: S. G. Dani
Journal: Proc. Amer. Math. Soc. 93 (1985), 67-72
MSC: Primary 28C10; Secondary 22D40, 43A05
DOI: https://doi.org/10.1090/S0002-9939-1985-0766529-3
MathSciNet review: 766529
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that under certain general conditions any finitely additive measure which is defined for all subsets of a set $ X$ and is invariant under the action of a group $ G$ acting on $ X$ is concentrated on a $ G$-invariant subset $ Y$ on which the $ G$-action factors to that of an amenable group. The result is then applied to prove a conjecture of $ S$. Wagon about finitely additive measures on spheres.


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

  • [1] S. G. Dani, On invariant finitely additive measures for automorphism groups of tori, Trans. Amer. Math. Soc. (to appear). MR 766213 (85m:28019)
  • [2] F. P. Greenleaf, Invariant means on topological groups, Van Nostrand Reinhold, New York, 1969. MR 0251549 (40:4776)
  • [3] J. Mycielski, Finitely additive invariant measures. I, Colloq. Math. 42 (1979), 309-318. MR 567569 (82g:43003a)
  • [4] J. Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), 623-636. MR 610970 (83a:28026)
  • [5] A. Tarski, Algebraische Fassung des Massproblems, Fund. Math. 31 (1938), 47-66.
  • [6] S. Wagon, Invariance properties of finitely additive measures on $ {{\mathbf{R}}^n}$, Illinois J. Math. 25 (1981), 74-86. MR 602898 (82c:28026)
  • [7] -, The use of shears to construct paradoxes in $ {{\mathbf{R}}^2}$, Proc. Amer. Math. Soc. 85 (1982), 353-359. MR 656101 (84d:28017)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28C10, 22D40, 43A05

Retrieve articles in all journals with MSC: 28C10, 22D40, 43A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0766529-3
Keywords: Invariant finitely additive measures, amenable groups
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society