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The structure of hyperfinite Borel equivalence relations

Dougherty, R. and Jackson, S. and Kechris, A. S. (1994) The structure of hyperfinite Borel equivalence relations. Transactions of the American Mathematical Society, 341 (1). pp. 193-225. ISSN 0002-9947.

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We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.

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Additional Information:© 1994 American Mathematical Society. Received by the editors September 24, 1991. Research of the authors was partially supported by NSF grants DMS-9158092 (R.D.), DMS-9007808 (S.J.) and DMS-9020153 (A.S.K.).
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MathSciNet review1149121
Issue or Number:1
Classification Code:1991 Mathematics Subject Classification: Primary 28D05; Secondary 03E15
Record Number:CaltechAUTHORS:20130521-092754518
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38593
Deposited By: Tony Diaz
Deposited On:21 May 2013 16:52
Last Modified:03 Oct 2019 04:58

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