The automorphism group of the Gaussian measure cannot act pointwise
E. Glasner, B. Tsirelson and B. Weiss,
"The automorphism group of the Gaussian measure cannot act pointwise."
Israel Journal of Mathematics 148, 305-329 (2005).
Available online (free of charge) from e-print archive (USA):
or its Israeli mirror:
A research paper, 25 pages, bibl. 34 refs.
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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