  We show that a {\it Borel} action of a Polish group on a standard Borel space
is Borel isomorphic to a {\it 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 $\Bbb N$, with the model-theoretic Vaught
Conjecture for infinitary logic; this depends on our identification of the
universal action for that group.

Logic;Functional Analysis