测度的集中与波兰群的旋转作用
Concentration of measure and whirly actions of Polish groups
摘要：
一个波兰群$G$在标准勒贝格测度空间$(X,μ)$上的弱连续近作用是涡动的，如果对$G$中的每一个严格正测度的$A\subseteq X$和每一个单位元的邻域$V$，集合$VA$有满测度.这是遍历性的一个强版本，而局部紧群从不承认旋转作用。相反，在Gromov和Milman意义下的波兰Lévy群（如$U(\ell^2)$）的每一个遍历近作用都是旋转的。我们给出了标准勒贝格测度空间（具有弱拓扑）的保测度自同构群$\Aut(X,μ)$的闭子群的例子，其重言式作用在$(X,μ)$上是whirly的，并且不是Lévy群，从而回答了Reynner和Weiss的一个问题.
英文摘要：
A weakly continuous near-action of a Polish group $G$ on a standard Lebesgue measure space $(X,μ)$ is whirly if for every $A\subseteq X$ of strictly positive measure and every neighbourhood $V$ of identity in $G$ the set $VA$ has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic near-action by a Polish Lévy group in the sense of Gromov and Milman, such as $U(\ell^2)$, is whirly (Glasner--Tsirelson--Weiss). We give examples of closed subgroups of the group $\Aut(X,μ)$ of measure preserving automorphisms of a standard Lebesgue measure space (with the weak topology) whose tautological action on $(X,μ)$ is whirly, and which are not Lévy groups, thus answering a question of Glasner and Weiss.
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