Dichotomy of ergodic measures on linear spaces
type: Item
instance of: scholarly article
title: Dichotomy of ergodic measures on linear spaces
zbMATH Open document ID: 0873.28012
DOI: 10.1007/BF02304900
author: S. G. Khaliullin
published in: Mathematical Notes
publication date: 1997-10-09T00:00:00Z
review text: The note generalizes the paper of \textit{Y. Okazaki} [Ann. Inst. H. Poincaré, Probab. Stat. 21, 393-400 (1985; Zbl 0582.60004)], which proved a dichotomy theorem for \(H\)-ergodic Hausdorff space equipped with a cylindrical \(\sigma\)-algebra \(C(E,x^*_i)\), on which a measure \(\mu\) is defined. In the present paper, the author does not assume that the linear space is equipped with a topological structure, and the measurable space structure need not agree with the linear operations. Let \((E,{\mathcal E})\) be a measure space such that \(E\) is a linear set and the \(\sigma\)-algebra \(\mathcal E\) is invariant under translations by elements \(x\in E\), and let \(\mu\) and \(\nu\) be two \(H\)-ergodic probability measures on \(\mathcal E\), where \(H\subset E\). Then \(\mu\) and \(\nu\) are either equivalent or mutually singular. This result has a generalization: \(\mu\) may be \(H_\mu\)-ergodic and \(\nu H_\nu\)-ergodic, where \(H_\mu\neq H_\nu\), but in this case \(H_\mu\) and \(H_\nu\) are linear subspaces of \(E\).
reviewed by: Victor Sharapov
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 60G30
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 921651
zbMATH Keywords: dichotomy
zbMATH Keywords: \(H\)-ergodic Hausdorff space
zbMATH Keywords: \(H\)-ergodic probability measures
MaRDI profile type: Publication
cites work: On equivalence of infinite product measures
cites work: Equivalence singularity dichotomies for a class of ergodic measures
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description: scientific article; zbMATH DE number 921651