BOREL ACTIONS OF POLISH GROUPS

We show that a Borel action of a Polish group on a standard Borel spac e 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 Bore l actions of Polish groups: universal actions, invariant probability m easures, and the Topological Vaught Conjecture. We establish the exist ence of universal actions for any given Polish group, extending a resu lt of Mackey and Varadarajan for the locally compact case. We prove an analog of Tarski's theorem on paradoxical decompositions by showing t hat the existence of an invariant Borel probability measure is equival ent to the nonexistence of paradoxical decompositions with countably m any Borel pieces. We show that various natural versions of the Topolog ical Vaught Conjecture are equivalent with each other and, in the case of the group of permutations of N, with the model-theoretic Vaught Co njecture for infinitary logic; this depends on our identification of t he universal action for that group.