Borel actions of Polish groups
- Creators
- Becker, Howard
- Kechris, Alexander S.
Abstract
We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures and the topological Vaught conjecture. We establish the existence of universal actions for any given Polish group, extending a result of Mackey and Varadarajan for the locally compact case. We prove an analogue of Tarski's theorem on paradoxical decompositions, by showing that the existence of an invariant Borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many Borel pieces. We show that various natural versions of the topological Vaught conjecture are equivalent to each other and, in the case of the group of permutations of N, with the model-theoretic Vaught conjecture for infinitary logic; this depends on our identification of the universal action for that group.
Additional Information
© 1993 American Mathematical Society. Received by the editors April 16, 1992 and, in revised form, October 15, 1992. The first author's research was partially supported by NSF Grant DMS-8914426. The second author's research was partially supported by NSF Grant DMS-9020153.Attached Files
Published - S0273-0979-1993-00383-5.pdf
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Additional details
- Eprint ID
- 38655
- Resolver ID
- CaltechAUTHORS:20130523-095229531
- DMS-8914426
- NSF
- DMS-9020153
- NSF
- Created
-
2013-05-29Created from EPrint's datestamp field
- Updated
-
2021-11-09Created from EPrint's last_modified field
- Other Numbering System Name
- MathSciNet Review
- Other Numbering System Identifier
- MR1185149