Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Uniqueness of invariant means for measure-preserving transformations

Author: Joseph Rosenblatt
Journal: Trans. Amer. Math. Soc. 265 (1981), 623-636
MSC: Primary 28D15; Secondary 43A07, 58F11
MathSciNet review: 610970
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For some compact abelian groups $ X$ (e.g. $ T^n$, $ n \geqslant 2$, and $ \prod\nolimits_{n = 1}^\infty {{Z_2}} $), the group $ G$ of topological automorphisms of $ X$ has the Haar integral as the unique $ G$-invariant mean on $ {L_\infty }(X,{\lambda _X})$. This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets $ \beta $ of $ {R^n}$, $ n \geqslant 3$; it is the unique normalized positive finitely-additive measure on $ \beta $ which is invariant under isometries and the transformation of $ {R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n})$. Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on $ {L_\infty }(X,\beta ,p)$, which is invariant by some group of measure-preserving transformations of the probability space $ (X,\beta ,p)$, are described.

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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28D15, 43A07, 58F11

Retrieve articles in all journals with MSC: 28D15, 43A07, 58F11

Additional Information

PII: S 0002-9947(1981)0610970-7
Article copyright: © Copyright 1981 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia