Open Access
VOL. 57 | 2010 Concentration of measure and whirly actions of Polish groups
Vladimir Pestov

Editor(s) Motoko Kotani, Masanori Hino, Takashi Kumagai

Adv. Stud. Pure Math., 2010: 383-403 (2010) DOI: 10.2969/aspm/05710383


A weakly continuous near-action of a Polish group $G$ on a standard Lebesgue measure space $(X, \mu)$ is whirly if for every $A \subseteq X$ of strictly positive measure and every neighbourhood $V$ of identity in $G$ the set $VA$ has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic near-action by a Polish Lévy group in the sense of Gromov and Milman, such as $U(\ell^2)$, is whirly (Glasner–Tsirelson–Weiss). We give examples of closed subgroups of the group Aut $(X, \mu)$ of measure preserving automorphisms of a standard Lebesgue measure space (with the weak topology) whose tautological action on $(X, \mu)$ is whirly, and which are not Lévy groups, thus answering a question of Glasner and Weiss.


Published: 1 January 2010
First available in Project Euclid: 24 November 2018

zbMATH: 1213.37011
MathSciNet: MR2648270

Digital Object Identifier: 10.2969/aspm/05710383

Primary: 37A15 , ‎37B05‎ , 43A05

Keywords: Concentration of measure on high-dimensional structures , greatest ambit , groups of measure-preserving transformations , Invariant measures , Lévy groups , Polish groups , whirly actions

Rights: Copyright © 2010 Mathematical Society of Japan


Back to Top