  For any standard Borel space $B$, let $\mathcal{P}(B)$ 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
$\mathbb{F}_2^\omega$, i.e., the measures in
$\mathcal{P}(B^{\mathbb{F}_2^\omega})$ that are invariant under the coordinate
permutations on $B^{\mathbb{F}_2^\omega}$ induced by all affine automorphisms
of $\mathbb{F}_2^{\omega}$. 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
$\mathcal{P}(B^{\mathbb{F}_2^\omega})$ is obtained from a
$\mathcal{P}(B)$-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
$\mathcal{P}(B^{\mathbb{F}_2^\omega})$ 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 $\mathbb{F}_2^n$ as $n\to\infty$. Via this correspondence, we
establish the above-mentioned group as a general limit domain valid for any
such sequence.
