The original formulation of de Finetti's theorem says that an exchangeable sequence of Bernoulli random variables is a mixture of iid sequences of random variables. Following the work of Hewitt and Savage, this theorem was previously known for several classes of exchangeable random variables (for instance, for Baire measurable random variables taking values in a compact Hausdorff space, and for Borel measurable random variables taking values in a Polish space). Under an assumption of the underlying common distribution being Radon, we show that de Finetti's theorem holds for a sequence of Borel measurable exchangeable random variables taking values in any Hausdorff space. This includes and generalizes the previously known versions of de Finetti's theorem. Furthermore, it shows that the theorem is explainable as a consequence of the nature of the distribution of the exchangeable random variables as opposed to the topological nature of their state space, which do not play a key role. Using known relative consistency results from set theory, this also shows that it is consistent with the axioms of ZFC that de Finetti's theorem holds for all sequences of exchangeable random variables taking values in any complete metric space. We use nonstandard analysis to first study the empirical measures induced by hyperfinitely many identically distributed random variables, which leads to a proof of de Finetti's theorem in great generality while retaining the combinatorial intuition of proofs of simpler versions of de Finetti's theorem. An overview of the requisite tools from nonstandard measure theory and topological meeasure theory is provided, with some new perspectives built at the interface between these fields as part of that overview. One highlight of this development is a new generalization of Prokhorov's theorem.

翻译：de Finetti定理的原始表述指出，一个可交换的伯努利随机变量序列是独立同分布随机变量序列的混合。继Hewitt与Savage的工作之后，该定理此前已对若干类可交换随机变量成立（例如，取值于紧致豪斯多夫空间中的贝尔可测随机变量，以及取值于波兰空间中的博雷尔可测随机变量）。在假定底层共同分布为拉东测度的条件下，我们证明de Finetti定理对取值于任意豪斯多夫空间中的博雷尔可测可交换随机变量序列成立。这包含了并推广了此前已知的de Finetti定理版本。进一步地，这表明该定理可解释为可交换随机变量分布性质的推论，而非其状态空间的拓扑性质——后者并不起关键作用。利用集合论中已知的相对一致性结果，我们还可证明：在ZFC公理下，de Finetti定理对取值于任意完备度量空间中的所有可交换随机变量序列一致地成立。我们采用非标准分析首先研究由超有限多个同分布随机变量导出的经验测度，从而在保留de Finetti定理简单版本证明的组合直觉的同时，给出了该定理在极大一般性下的证明。文中提供了非标准测度论与拓扑测度论所需工具的总览，并在这些领域的交叉处构建了一些新视角。此发展的一大亮点是Prokhorov定理的一个新推广。