The automorphism group of the Gaussian measure cannot act pointwise

By Eli Glasner, Boris Tsirelson, Benjamin Weiss. 2003

63 citations recorded by the source corpus.

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 BooleanG-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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91c81fe74504ea489f460575ceba7747dbbd3256