In order to understand the structure of the “typical” element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood; see, e.g., the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For Homeo + ([0, 1]) we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for Homeo + (S 1 ) we describe the non-Haar null conjugacy classes, and for U ( l 2 ) we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set.

中文翻译：

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随机同胚的结构

为了理解同胚群的“典型”元素的结构，必须研究该群的共轭类有多大。当典型意味着 Baire 类别意义上的通用时，这是很好理解的；参见例如 Glasner 和 Weiss 以及 Kechris 和 Rosendal 的作品。继 Dougherty 和 Mycielski 之后，我们使用 Christensen 的 Haar 零集概念研究了这个问题的测度理论对偶。当典型意味着随机时，即几乎所有关于 Haar 空集的概念，同胚的行为与泛型情况完全不同。对于 Homeo + ([0, 1]) 我们描述了非 Haar null 共轭类并且还表明它们的并集是 co-Haar null，对于 Homeo + (S 1 ) 我们描述了非 Haar null 共轭类，对于U ( l 2 ) 我们证明，除了多班次的类外，所有共轭类都是 Haar null。作为一个应用，我们肯定地回答了这些群是否可以写成一个微弱的和一个 Haar 空集的并集的问题。