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Ergodic measures are of weak product type

Published online by Cambridge University Press:  24 October 2008

Gavin Brown  and
A. H. Dooley
Gavin Brown
Affiliation:
School of Mathematics, the University of New South Wales, Kensington, Australia.
A. H. Dooley
Affiliation:
School of Mathematics, the University of New South Wales, Kensington, Australia.
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Abstract

There are many results which discuss ergodicity in terms of approximate product properties. Here we work throughout with stochastically independent σ-algebras (ℰ, ℱ are independent of μ if μ (E ∩ F) =μ (E)μ(F) for all E ∊ ℰ,F ∊ ℱ), to obtain an exact product property characteristic of (a large class of) ergodic measures. The ideas are based on work of A. V. Skorokhod on admissible translates of probabilities on Hilbert space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 1 , July 1985 , pp. 129 - 145
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Billingsley, P.. Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
[2]Bourbaki, N.. Eléments de mathématique. Livre VI, ch. 6. Intégration vectorielle (Hermann, 1959).Google Scholar
[3]Dye, H. A.. On groups of measure preserving transformations: II. Amer. J. Math. 85 (1963), 551–576.CrossRefGoogle Scholar
[4]Effros, E. G.. Polish transformation groups and classification problems. General Topology and Modern Analysis (Proc. Conf. Univ. California, Riverside, 1980), pp. 405–416 (Academic Press, 1981).Google Scholar
[5]Krieger, W.. On the infinite product construction of non-singular transformations of measure space. Inventiones Math. 15 (1972), 144–163.CrossRefGoogle Scholar
[6]Kuratowski, K.. Topology, vol. 1 (Academic Press, 1966).Google Scholar
[7]Loève, M.. Probability Theory, vol. II (Van Nostrand, 1960).Google Scholar
[8]Moore, C. C.. Ergodicity and type I groups. In L. Auslander and C. C. Moore, Unitary representations of solvable Lie groups. Mem. Amer. Math. Soc. 62 (1966).Google Scholar
[9]Powers, R. T.. Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. Math. 86 (1967), 138–171.CrossRefGoogle Scholar
[10]Skorokhod, A. V.. Integration in Hilbert Space (Springer-Verlag, 1974).CrossRefGoogle Scholar