## Borel actions of Polish groups

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- by Howard Becker and Alexander S. Kechris PDF
- Bull. Amer. Math. Soc.
**28**(1993), 334-341 Request permission

## 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 analog 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 with each other and, in the case of the group of permutations of ${\mathbb {N}}$, with the model-theoretic Vaught Conjecture for infinitary logic; this depends on our identification of the universal action for that group.

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## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**28**(1993), 334-341 - MSC: Primary 03E15; Secondary 03C75, 28E99, 54E99, 54H11
- DOI: https://doi.org/10.1090/S0273-0979-1993-00383-5
- MathSciNet review: 1185149