论文标题
不可数的Mackey-Zimmer定理
An uncountable Mackey-Zimmer theorem
论文作者
论文摘要
Mackey-Zimmer定理分类了Ergodic grout扩展名$ x $由紧凑型组$ k $ y $ y $的$ y $，通过表明此类扩展对某些$ k $ o $ k $ h $的封闭子组$ h $ y of skew-equiv-equiv-equiv y equiv y equiv y equiv y equiv y \ equiv y \ equiv y \ equiv y \rtimes_ρh$。也可以使用类似的定理，可用于$ y $ $ y $ x $ x $ y $ x $ x $ y $的类似定理，即它们与同质偏斜产品$ y \ y \rtimes_ρh/m $同构。这些定理在千古理论中有许多用途，例如在host-kra kra结构理论中起关键作用的特征因素的特征性因素。 Mackey-Zimmer定理的现有证明需要各种“可视性”，“可分离性”或“ Metrizanity”假设，该假设是在系统上，基本空间$ y $以及用于执行扩展程序的组$ k $的组$γ$。在本文中，我们将Mackey-Zimmer定理概括为“不可数”的设置，在该设置中，这些假设被省略了，以制定具有量度的保留系统的概念，而组扩展更加抽象。但是，通过使用“规范模型”对伴侣论文中开发的抽象测量系统的使用“规范模型”部分抵消了这种抽象。在随后的工作中，我们将应用该定理以获得宿主-KRA结构理论的不可数版本。
The Mackey-Zimmer theorem classifies ergodic group extensions $X$ of a measure-preserving system $Y$ by a compact group $K$, by showing that such extensions are isomorphic to a group skew-product $X \equiv Y \rtimes_ρH$ for some closed subgroup $H$ of $K$. An analogous theorem is also available for ergodic homogeneous extensions $X$ of $Y$, namely that they are isomorphic to a homogeneous skew-product $Y \rtimes_ρH/M$. These theorems have many uses in ergodic theory, for instance playing a key role in the Host-Kra structural theory of characteristic factors of measure-preserving systems. The existing proofs of the Mackey-Zimmer theorem require various "countability", "separability", or "metrizability" hypotheses on the group $Γ$ that acts on the system, the base space $Y$, and the group $K$ used to perform the extension. In this paper we generalize the Mackey-Zimmer theorem to "uncountable" settings in which these hypotheses are omitted, at the cost of making the notion of a measure-preserving system and a group extension more abstract. However, this abstraction is partially counteracted by the use of a "canonical model" for abstract measure-preserving systems developed in a companion paper. In subsequent work we will apply this theorem to also obtain uncountable versions of the Host-Kra structural theory.
