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On the existence of cocycle-invariant Borel probability measures

Part of: Set theory Classical measure theory

Published online by Cambridge University Press:  12 April 2019

BENJAMIN D. MILLER Open the ORCID record for BENJAMIN D. MILLER [Opens in a new window]
BENJAMIN D. MILLER*
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090Wien, Austria email benjamin.miller@univie.ac.at
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Abstract

We show that a natural generalization of compressibility is the sole obstruction to the existence of a cocycle-invariant Borel probability measure.

Keywords

Cocyclecompressioninvariant measure

MSC classification

Primary: 03E15: Descriptive set theory 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 3150 - 3168
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Copyright
© Cambridge University Press, 2019

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References

Becker, H. and Kechris, A. S.. The Descriptive Set Theory of Polish Group Actions (London Mathematical Society Lecture Note Series, 232) . Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
Dougherty, R., Jackson, S. and Kechris, A. S.. The structure of hyperfinite Borel equivalence relations. Trans. Amer. Math. Soc. 341(1) (1994), 193–225.CrossRefGoogle Scholar
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
Hopf, E.. Theory of measure and invariant integrals. Trans. Amer. Math. Soc. 34(2) (1932), 373–393.CrossRefGoogle Scholar
Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156) . Springer, New York, 1995.CrossRefGoogle Scholar
Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence (Lecture Notes in Mathematics, 1852) . Springer, Berlin, 2004.CrossRefGoogle Scholar
Kechris, A. S., Solecki, S. and Todorcevic, S.. Borel chromatic numbers. Adv. Math. 141(1) (1999), 1–44.CrossRefGoogle Scholar
Miller, B. D.. The existence of measures of a given cocycle. II. Probability measures. Ergod. Th. & Dynam. Sys. 28(5) (2008), 1615–1633.CrossRefGoogle Scholar
Miller, B.. The existence of measures of a given cocycle. I. Atomless, ergodic 𝜎-finite measures. Ergod. Th. & Dynam. Sys. 28(5) (2008), 1599–1613.CrossRefGoogle Scholar
Murray, F. J. and Von Neumann, J.. On rings of operators. Ann. of Math. (2) 37(1) (1936), 116–229.CrossRefGoogle Scholar
Nadkarni, M. G.. On the existence of a finite invariant measure. Proc. Indian Acad. Sci. Math. Sci. 100(3) (1990), 203–220.CrossRefGoogle Scholar
Weiss, B.. Measurable dynamics. Conference in Modern Analysis and Probability (New Haven, Conn., 1982) (Contemporary Mathematics, 26) . American Mathematical Society, Providence, RI, 1984, pp. 395–421.CrossRefGoogle Scholar