Countable sections for free actions of groups
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.
scientific article; zbMATH DE number 3977333
scientific article