On Fω2-affine-exchangeable probability measures

Pablo, Candela and Gonzalez Sanchez, Diego and Szegedy, Balázs (2024) On Fω2-affine-exchangeable probability measures. STUDIA MATHEMATICA. ISSN 0039-3223 (In Press)

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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 Fω2, i.e., the measures in P(BFω2) that are invariant under the coordinate permutations on BFω2 induced by all affine automorphisms of Fω2. 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 P(BFω2) 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(BFω2) 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 Fn2 as n→∞. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.

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