The existence of invariant probability measures for a group of transformations
type: Item
instance of: scholarly article
title: The existence of invariant probability measures for a group of transformations
zbMATH Open document ID: 0786.28012
DOI: 10.1007/BF02784061
author: P. Zakrzewski
published in: Israel Journal of Mathematics
publication date: 1993-12-07T00:00:00Z
review text: Let \(G\) be a group of measurable transformations of a \(\sigma\)-finite measure space \((X,{\mathcal A},m)\). A set \(A\in{\mathcal A}\) is said to be weakly wandering under \(G\) if there exists a sequence \(g_ n\) of elements of \(G\) such that the sets \(g_ n A\) are all pairwise disjoint. The author proves that the following conditions are equivalent: (i) There exists a \(G\)-invariant probability measure \(\mu\) defined on \(\mathcal A\) such that the measure \(m\) is absolutely continuous with respect to \(\mu\). (ii) There does not exist any set of positive measure which is weakly wandering under \(G\). This result gives a proof of a conjecture of Chuaqui. Under additional assumptions, versions of this result were known before; in particular when \(G\) is cyclic (due to Hopf), and when \(G\) is an arbitrary group of nonsingular transformations (Hajian and Itô).
Mathematics Subject Classification ID: 28D15
zbMATH DE Number: 446879
zbMATH Keywords: groups of measure-preserving transformations
zbMATH Keywords: invariant probability measure
zbMATH Keywords: weakly wandering
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 446879