Generalizing the de Finetti--Hewitt--Savage theorem

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.