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 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.
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Glasner, E., Tsirelson, B. & Weiss, B. The automorphism group of the Gaussian measure cannot act pointwise. Isr. J. Math. 148, 305–329 (2005). https://doi.org/10.1007/BF02775441
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DOI: https://doi.org/10.1007/BF02775441