The automorphism group of the Gaussian measure cannot ACT pointwise

E. Glasner*, B. Tsirelson, B. Weiss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)305-329
Number of pages25
JournalIsrael Journal of Mathematics
StatePublished - 2005


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