Random metric spaces and the universal Urysohn space.2
Random metric spaces and the universal Urysohn space.2
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随机度量空间和通用 Urysohn 空间.2
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发表时间：
2002
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通讯作者：
A. Vershik
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作者：
A. Vershik
我们引入了所有波兰（=可分离完全度量）空间集合的模型，即距离矩阵的锥体 R，并考虑与该对象相关的几何和概率问题。我们证明了该模型意义上的通用波兰空间是所谓的通用 Urysohn 空间，由 P.S.Urysohn 在 1920 年代定义。然后我们考虑具有度量（度量三元组）的度量空间并定义其矩阵分布的完整不变量。我们给出了矩阵分布的内在表征，并使用遍历定理给出了格罗莫夫重构定理的新证明。给出了圆锥 R 上一类广泛测度的自然构造，并且对于这些测度，我们证明随机波兰空间再次是 Urysohn 空间的概率为 1。这些问题与多个参数的可测量函数的度量分类和无限对称群的作用分类（[4, 8]）有着紧密的联系。接下来将介绍度量空间统计理论的应用。
We introduce a model of the set of all Polish (=separable complete metric) spaces which is the cone R of distance matrices, and consider the geometrical and probabilistic problems connected with this object. We prove that the generic Polish space in the sense of this model is the so called universal Urysohn space which was defined by P.S.Urysohn in the 1920-th. Then we consider the metric spaces with measures (metric triples) and define a complete invariant of its-matrix distribution. We give an intrinsic characterization of matrix distribution and using the ergodic theorem give a new proof of Gromov's reconstruction theorem. A natural construction of a wide class of measures on the cone R is given and for these we show that with probability one the random Polish space is again the Urysohn space. There is a tight link of these questions with metric classification of measurable functions of several arguments and classification of the actions of infinite symmetric group ([4, 8]) Applications to the statistical theory of metric space will follow.