Spatial models of Boolean actions and groups of isometries.

**Abstract.** I will present a result showing that each measure preserving Boolean
action of a Polish group of isometries of a locally compact separable metric space
has a spatial model or, in other words, has a point realization. This result extends
a classical theorem of Mackey and a recent theorem of Glasner and Weiss. The proof
of the result requires a new characterization of Polish groups of isometries of
locally compact separable metric spaces which may be of independent
interest, and which I will also present. The solution to Hilbert's fifth problem
plays an important role in establishing this characterization. This is a joint work
with Aleksandra Kwiatkowska.