A sequence of random variables is called exchangeable if its joint distribution is invariant under permutations. The original formulation of de Finetti's theorem says that any exchangeable sequence of $\{0,1\}$-valued random variables can be thought of as a mixture of independent and identically distributed sequences in a certain precise mathematical sense. Interpreting this statement from a convex analytic perspective, Hewitt and Savage obtained the same conclusion for more general state spaces under some topological conditions. The main contribution of this paper is in providing a new framework that explains the theorem purely as a consequence of the underlying distribution of the random variables, with no topological conditions (beyond Hausdorffness) on the state space being necessary if the distribution is Radon. We also show 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. The framework we use is based on nonstandard analysis. We have provided a self-contained introduction to nonstandard analysis as an appendix, thus rendering measure theoretic probability and point-set topology as the only prerequisites for this paper. Our introduction aims to develop some new ideologies that might be of interest to mathematicians, philosophers, and mathematics educators alike. Our technical tools come from nonstandard topological measure theory, in which a highlight is a new generalization of Prokhorov's theorem. Modulo such technical tools, our proof relies on properties of the empirical measures induced by hyperfinitely many identically distributed random variables -- a feature that allows us to establish de Finetti's theorem in the generality that we seek while still retaining the combinatorial intuition of proofs of simpler versions of de Finetti's theorem.

翻译：随机变量序列被称为可交换的，若其联合分布在置换下保持不变。德·芬内蒂定理的原始表述指出，任何取值为$\{0,1\}$的可交换随机变量序列，在精确的数学意义上可视为独立同分布序列的混合。从凸分析的角度解读这一陈述，休伊特和萨维奇在一定的拓扑条件下，对更一般的状态空间得到了相同的结论。本文的主要贡献在于提供一个全新框架，该框架纯粹基于随机变量的潜在分布来解释该定理，且若分布为拉东测度，则状态空间无需拓扑条件（除豪斯多夫性外）。我们还证明，在ZFC公理体系下，德·芬内蒂定理对任何取值为完备度量空间的可交换随机变量序列均成立。我们所使用的框架基于非标准分析。我们在附录中提供了非标准分析的独立导引，因此本文仅需测度论概率论和点集拓扑学作为先修知识。此导引旨在发展一些可能引起数学家、哲学家及数学教育工作者兴趣的新理念。我们的技术工具源自非标准拓扑测度论，其中亮点之一是普罗霍罗夫定理的新推广。借助这类技术工具，我们的证明依赖于由超有限多个独立同分布随机变量导出的经验测度性质——这一特性使我们能够在保留德·芬内蒂定理简单版本证明的组合直觉的同时，建立其所需的广义形式。