Measure-theoretic problems in topological dynamics
type: Item
instance of: scholarly article
title: Measure-theoretic problems in topological dynamics
zbMATH Open document ID: 0870.28009
DOI: 10.1007/BF02788772
author: S. Grekas
published in: Journal d'Analyse Mathématique
publication date: 1995-11-16T00:00:00Z
review text: In this interesting paper, the author is concerned with point realizations of \(\sigma\)-homomorphisms, with compact topological transformation groups, and with distal flows. In the principal Theorem 3.7, it is shown that the phase space \(X\) of a minimal distal flow \((X,T)\) is Baire isomorphic to a product of compact metric spaces, and the isomorphism takes some invariant completion-regular probability measure to a direct product measure. This theorem is a significant advance in the literature, and has significant consequences. For instance: If \(\nu\) is any finite Baire measure on \(X\) (as above), then every automorphisms of the measure algebra of \(\nu\) is induced by an invertible completion Baire point transformation of \(X\). If \(\lambda\), \(\nu\) are Radon probability measures on \(X\) as above, and if \(\exists\) a measure-preserving isomorphism of their measure algebras, then \(\lambda\) and \(\nu\) are completion Baire isomorphic. Correction: The author of reference [23] in the bibliography (p. 220) is D. McMahon.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 28A51
zbMATH DE Number: 807407
zbMATH Keywords: point realizations
zbMATH Keywords: compact topological transformation groups
zbMATH Keywords: distal flows
zbMATH Keywords: completion-regular probability measure
zbMATH Keywords: invertible completion Baire point transformation
zbMATH Keywords: measure-preserving isomorphism
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 807407