Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-29T01:12:52.513Z Has data issue: false hasContentIssue false

Finitely-additive invariant measures on Euclidean spaces

Published online by Cambridge University Press:  19 September 2008

Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that for n ≥ 3 the Lebesgue measure is the unique finitely-additive isometry-invariant measure on the ring of bounded Lebesgue measurable subsets of the n-dimensional Euclidean space.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Banach, S.. Sur le problème de la mesure. In S. Banach Oeuvres, vol. 1, Warsaw, 1967, pp. 318322.Google Scholar
[2]Borel, A.. Some finiteness properties of Adele groups over number fields. Publ. Math. I.H.E.S. 16 (1963), 130.Google Scholar
[3]Bourbaki, N.. Éléments de mathematique, Première partie, Livre VI, Integration. Hermann: Paris.Google Scholar
[4]Delaroche, C. & Kirillov, A.. Sur les relation entre l'espace d'une groupe et la structure de ses sous-groupes fermes. Séminaire Bourbaki, Exposé 343, 1968. Lecture Notes in Math. No. 180. Springer: Berlin-Heidelberg-New York, 1972.Google Scholar
[5]Del Junco, A. & Rosenblatt, J.. Counter examples in ergodic theory and number theory. Math. Ann. 245 (1979) 185197.Google Scholar
[6]Kazhdan, D.. On a connection of the dual space of the group with the structure of its closed subgroups. Fund. Anal. Prilozhen 1 (1967) 7174. (In Russian). (Math. Rev. 35 288.)Google Scholar
[7]Losert, V. & Rindler, H.. Almost invariant sets. Bull. London Math. Soc. (To appear.)Google Scholar
[8]Mackey, G.. Unitary representation of group extensions I. Ada Math. 99 (1958) 265311.Google Scholar
[9]Margulis, G.. Some remarks on invariant means. Mh. Math. 90 (1980) 233235.Google Scholar
[10]Prasad, G.. Triviality of certain automorphisms of sime-simple groups over local fields. Math. Ann. 218 219227.Google Scholar
[11]Rosenblatt, J.. Uniqueness of invariant means for measure-preserving transformations. Trans. Amer. Math. Soc. (To appear.)Google Scholar
[12]Schmidt, K.. Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic actions. Preprint.Google Scholar
[13]Sullivan, D.. For n ≥ 3 there is only one finitely-additive rotationally-invariant measure on the n-sphere defined on all Lebesgue measurable sets. Bull. Amer. Math. Soc. 4 No. 1 (1980).Google Scholar