Countable sections for free actions of groups
type: Item
instance of: scholarly article
title: Countable sections for free actions of groups
zbMATH Open document ID: 0605.28009
DOI: 10.1016/0001-8708(85)90091-X
author: B. George
published in: Advances in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Consider an analytic measurable group (G,\({\mathcal B})\) acting freely and measurably on an analytic measurable space (X,\({\mathcal A})\). The authors prove that if (X,\({\mathcal A})\) admits a quasi-invariant measure \(\mu\) and a countable measurable section Y such that \(\mu (GY)>0\) then (G,\({\mathcal B})\) admits a quasi-invariant probability measure; we recall here that a subset Y is called a countable section if \(Gx\cap Y\) is countable for all \(x\in X.\) By a well-known theorem of G. W. Mackey the conclusion also implies that G admits the structure of a locally compact group with \({\mathcal B}\) as the \(\sigma\)-algebra of Borel sets. The result may thus be viewed as a converse of a theorem of \textit{P. Forrest} on the existence of countable measurable sections Y, such that \(\mu (GY)=1,\) for actions of locally compact groups, in the presence of a quasi-invariant measure \(\mu\) [cf. Adv. Math. 14, 271-308 (1974; Zbl 0288.28021)]. It is deduced in particular, that if \(X=\prod^{\infty}_{1}{\mathbb{T}},\) the infinite direct product of circle groups and \(G=\oplus^{\infty}_{1}{\mathbb{T}},\) the infinite direct sum, then the action of G on X is not hyperfinite. Though this is stated in a paper of \textit{J. Feldman}, \textit{P. Hahn} and \textit{C. C. Moore} [Adv. Math. 28, 186-230 (1978; Zbl 0392.28023)]. A. Connes is said to have pointed out that the proof there is incomplete.
Mathematics Subject Classification ID: 28C10
Mathematics Subject Classification ID: 28D99
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 3977333
zbMATH Keywords: analytic measurable group
zbMATH Keywords: quasi-invariant measure
zbMATH Keywords: countable measurable section
MaRDI profile type: Publication
cites work: Orbit structure and countable sections for actions of continuous groups
cites work: Borel Structure in Groups and Their Duals
cites work: Ergodic theory and virtual groups
cites work: On rings of operators. Reduction theory
full work available at URL: 0001-8708(85)90091-x
OpenAlex ID: W2067916929
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description: scientific article; zbMATH DE number 3977333