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.
