de Finetti–Hewitt–Savage定理的推广
Generalizing the de Finetti--Hewitt--Savage theorem
AI总结：
本研究利用非标准分析工具，将de Finetti–Hewitt–Savage定理推广至任意豪斯多夫空间上的Radon分布可交换序列，还给出非标准分析入门介绍及Prokhorov定理的新推广。
摘要：
若一个随机变量序列的联合分布在指标置换下保持不变，则称该序列是“可交换的”。de Finetti定理的最初表述大致为：任何取值于{0,1}的可交换随机变量序列，都可视为独立同分布序列的混合。Hewitt和Savage证明，在某些拓扑条件下，取值于更一般状态空间的可交换随机变量序列也满足相同结论。\n 我们利用非标准分析工具证明：取值于任意豪斯多夫状态空间、服从Radon分布的可交换随机变量序列，一定可以表示为独立同分布随机变量序列的混合。\n 本工作的呈现采用面向广大研究生层次数学读者的“讲义”风格——正文主体以基于历史背景的问题介绍开篇，预示了我们将通过一系列附录展开的技术方法。在引言之后的简短章节中，我们运用这些技术给出了主要结果的自足证明。\n 我们在第一个附录中提供了自足的、带有哲学动机的非标准分析介绍，因此本工作的先修要求仅为测度论概率论和点集拓扑的入门课程。该介绍旨在发展关于这一主题的一些新思想，可能会引起数学家、哲学家和数学教育工作者的兴趣。其余附录的一大亮点是在任意豪斯多夫空间上的所有概率测度空间的框架下，对Prokhorov定理的新推广。
英文摘要：
A sequence of random variables is called \textit{exchangeable} if its joint distribution is invariant under permutations of indices. The original formulation of de Finetti's theorem roughly says that any exchangeable sequence of $\{0,1\}$-valued random variables can be thought of as a mixture of independent and identically distributed sequences. Hewitt and Savage were able to obtain the same conclusion for exchangeable sequences of random variables taking values in more general state spaces under some topological conditions. Using tools from nonstandard analysis we prove that an exchangeable sequence of Radon-distributed random variables taking values in any Hausdorff state space must be representable as a mixture of sequences of independent and identically distributed random variables. Our presentation of this work follows the style of \textit{lecture notes} intended for broad graduate-level mathematical audiences -- the main body of the manuscript starts with a historically grounded introduction to the problem, foreshadowing our techniques that are developed via a series of appendices. These techniques are used to provide self-contained proofs of our main results in a short section following the introduction. We have provided a self-contained philosophically motivated introduction to nonstandard analysis in the first appendix, thus rendering first courses in measure theoretic probability and point-set topology as the only prerequisites for the work. This introduction aims to develop some new ideologies about the subject that might be of interest to mathematicians, philosophers, and mathematics educators alike. One highlight of the rest of the appendices is a new generalization of Prokhorov's theorem in the setting of the space of all probability measures on arbitrary Hausdorff spaces.
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