The automorphism group of the Gaussian measure cannot act pointwise

Details The automorphism group of the Gaussian measure cannot act pointwise The automorphism group of the Gaussian measure cannot act pointwise

2003-11-25

arxiv.org/abs/math/0311450v2

Israel Journal of Mathematics 148 (2005), 305-329.

Mathematics

Dynamical Systems

v2 (final), 18 pages. Added: remarks 1.4, 1.5, 1.9, 3.6 (supplemented); Appendix B; some historical background; refs 1, 5, 12,

Scientific paper

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.

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