## B. Tsirelson | ## The automorphism group of the Gaussian measure cannot act pointwise | ## Recent works |

E. Glasner, B. Tsirelson and B. Weiss,

"The automorphism group of the Gaussian measure cannot act
pointwise."

math.DS/0311450.

Israel Journal of Mathematics **148**, 305-329 (2005).

Available online (free of charge) from e-print archive (USA):

arXiv.org/abs/math.DS/0311450/

or its Israeli mirror:

il.arXiv.org/abs/math.DS/0311450/

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.

- Introduction.
- Levy groups admit no spatial actions.
- Which actions admit spatial models?
- Whirly actions.
- The automorphism group of the Gaussian measure.
- Appendices A,B.

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