2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/97900Classical 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.v2 (final), 18 pages. Added: remarks 1.4, 1.5, 1.9, 3.6 (supplemented); Appendix B; some historical background; refs 1, 5, 12, 18, 20-25, 29, 32Dynamical SystemsProbability28D15; 28C20, 60A10, 60G15, 37B05text