Generalizing the de Finetti--Hewitt--Savage theorem

Irfan Alam

Published 2020-08-20Version 1

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 is 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 currently known versions of de Finetti's theorem. 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. The required tools from topological measure theory are developed with the aid of perspectives provided by nonstandard measure theory. One highlight of this development is a new generalization of Prokhorov's theorem.