On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

Abstract

For any standard Borel space B, let P(B) denote the space of Borel probability measures on B. 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 ω 2 , i.e., the measures in P(B F ω 2 ) that are invariant under the coordinate permutations on B F ω 2 induced by all affine automorphisms of F ω 2 . 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 affineexchangeable measure in P(B F ω 2 ) is obtained from a P(B)-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 P(B F ω 2 ) equipped with the vague topology (when B 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 n 2 as n → ∞. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.