The automorphism group of the Gaussian measure cannot act pointwise
Dynamical Systems
2007-05-23 v2 Probability
Abstract
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.
Cite
@article{arxiv.math/0311450,
title = {The automorphism group of the Gaussian measure cannot act pointwise},
author = {E. Glasner and B. Tsirelson and B. Weiss},
journal= {arXiv preprint arXiv:math/0311450},
year = {2007}
}
Comments
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, 32