  Classical ergodic theory deals with measure (or measure class) preserving
actions of locally compact groups on Lebesgue spaces. An important tool in this
setting is a theorem of Mackey which provides spatial models for Boolean
G-actions. We show that in full generality this theorem does not hold for
actions of Polish groups. In particular there is no Borel model for the Polish
automorphism group of a Gaussian measure. In fact, we show that this group as
well as many other Polish groups do not admit any nontrivial Borel measure
preserving actions.
