A Borel parametrization of Polish groups
A Borel parametrization of Polish groups
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波兰群的 Borel 参数化
DOI：
10.2977/prims/1195178509
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发表时间：
1985
影响因子：
1.2
通讯作者：
Colin E. Sutherland
中科院分区：
文献类型：
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作者：
Colin E. Sutherland
本文构造了一个标准的Borel空间Pg和一个映射p^Pg->；G(P)，使得每个G(P)都是Polish群，并且使得每个Polish群至少同构于G(P)中的一个，从而Pg可作为所有Polish群的参数空间。我们建立了从标准B空间到Polish群的Borel映射和从标准Borel群胚到Polish群的Borel函子的概念；这两个概念都是通过PG的Borel分解的存在性来定义的。我们应用这些思想建立了一个一般的“上同调引理”，断言当基础群胚是一个超有限等价关系时，值在波兰群的Borel族中的上循环可以上有界到一个给定的稠密、正规的Borel子群族中。本文的目的是通过标准Borel空间PG给出Polish拓扑群空间的一种参数化，即那些其基本拓扑可以由完备度量定义的第二可数拓扑群，并给出它在从标准Borel群胚到Polish群的“Borel函子”概念中的应用。在与M.Takesaki共同研究超有限、半有限内射von Neumann代数上离散顺从群的可能作用(直到余圈共轭)的过程中，对这些概念的需求变得明显[12]，这篇论文可以看作是这项工作的准备工作。然而，所采用的观点也揭示了A.Connes[4]将Borel函子的概念从标准Borel群胚到标准测度空间的定义是非常自然的。这里考虑的问题出现的一种常见情况是：如果G是波兰群，X是波兰G-空间，则
This paper constructs a standard Borel space, PG, and a map p^PG—>G(p} such that each G(p) is a Polish group, and such that every Polish group is isomorphic to at least one of the groups G(p) ; PG thus serves as a parameter space for all Polish groups. We formulate the notion of a Borel map from a standard B Space to Polish groups, and that of a Borel functor from a standard Borel groupoid to Polish groups; both are defined in terms of the existence of Borel factorizations through PG. We apply these ideas to establish a general "Cohomology Lemma," asserting that cocycles, with values in Borel family of Polish groups, may be cobounded into a given family of dense, normal, Borel subgroups, whenever the underlying groupoid is a hyperfinite equivalence relation. The purpose of this paper is to provide a parametrizatio n, by a standard Borel space PG, for the space of Polish topological groups, i. e. those second countable topological groups whose underlying topology may be defined by a complete metric, and to present applications of this to the notion of "Borel functor" from a standard Borel groupoid to Polish groups. The need for such concepts became apparent during the course of joint work with M. Takesaki on the classification of the possible actions (up to cocycle conjugacy) of a discrete amenable group on a hyperfinite, semifinite injective von Neumann algebra [12], and the paper can be viewed as preparatory to this work. However, the point of view adopted also reveals a definition of A. Connes, [4], of the notion of Borel functor from a standard Borel groupoid to standard measure spaces, as being very natural. A common situation in which the problems considered here arise is the following: if G is a Polish group and X a Polish G-space under