The automorphism group of the Gaussian measure cannot act pointwise
The automorphism group of the Gaussian measure cannot act pointwise
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高斯测度的自同构群不能逐点作用
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
Benjamin Weiss
中科院分区：
文献类型：
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作者：
E. Glasner;B. Tsirelson;Benjamin Weiss
经典遍历理论研究Lebesgue空间上局部紧群的测度（或测度类）保持作用。在这种情况下的一个重要工具是麦基定理，它提供了空间模型的布尔G-行动。我们表明，在充分的一般性，这一定理并不适用于波兰集团的行动。特别是没有波莱尔模型的波兰自同构群的高斯措施。事实上，我们表明，这个群体以及许多其他波兰团体不承认任何非平凡的波莱尔措施保持行动。
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.