On F ω 2-affine-exchangeable probability measures

For any standard Borel space , let denote the space of Borel probability measures on . In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space F, i.e., the measures in F that are invariant under the coordinate permutations on F induced by all affine automorphisms of F. We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affine-exchangeable measure in F is obtained from a -valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in F equipped with the vague topology (when is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces F as . Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.