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The existence of measures of a given cocycle, I: atomless, ergodic σ-finite measures

Published online by Cambridge University Press:  01 October 2008

BENJAMIN MILLER
BENJAMIN MILLER*
Affiliation:
UCLA Department of Mathematics, 520 Portola Plaza, Los Angeles, CA 90095-1555, USA (email: bdm@math.ucla.edu)
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Abstract

Given a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle , we characterize the circumstances under which there is a suitably non-trivial σ-finite measure μ on X such that, for every Borel injection ϕ whose graph is contained in E, ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1599 - 1613
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.CrossRefGoogle Scholar
[2]Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156). Springer, New York, 1995.CrossRefGoogle Scholar
[3]Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence (Lecture Notes in Mathematics, 1852). Springer, Berlin, 2004.CrossRefGoogle Scholar
[4]Kechris, A. S., Solecki, S. and Todorčević, S.. Borel chromatic numbers. Adv. Math. 141(1) (1999), 1–44.CrossRefGoogle Scholar
[5]Rudin, W.. Real and Complex Analysis. McGraw-Hill, New York, 1987.Google Scholar
[6]Schmidt, K.. Cocycles on Ergodic Transformation Groups (Macmillan Lectures in Mathematics, 1). Macmillan Company of India, Ltd., Delhi, 1977.Google Scholar
[7]Shelah, S.. Can you take Solovay’s inaccessible away? Israel J. Math. 48(1) (1984), 1–47.CrossRefGoogle Scholar
[8]Slaman, T. and Steel, J. R.. Definable Functions on Degrees (Cabal Seminar, Lecture Notes in Mathematics, 1333). Eds. A. S. Kechris, D. A. Martin and J. R. Steel. Springer, Berlin, 1988, pp. 81–85.Google Scholar
[9]Shelah, S. and Weiss, B.. Measurable recurrence and quasi-invariant measures. Israel J. Math. 43(2) (1982), 154–160.CrossRefGoogle Scholar
[10]Weiss, B.. Measurable dynamics. Contemp. Math. 26 (1984), 395–421.CrossRefGoogle Scholar