On the geometry of a class of invariant measures and a problem of Aldous

In his survey [4] of notions of exchangeability, Aldous introduced a form of exchangeability corresponding to the symmetries of the infinite discrete cube, and asked whether these exchangeable probability measures enjoy a representation theorem similar to those for exchangeable sequences [11], arrays [12, 13, 1, 2] and set-indexed families [15]. In this note we to prove that, whereas the known representation theorems for different classes of partially exchangeable probability measure imply that the compact convex set of such measures is a Bauer simplex (that is, its subset of extreme points is closed), in the case of cube-exchangeability it is a copy of the Poulsen simplex (in which the extreme points are dense). This follows from the arguments used by Glasner and Weiss’ for their characterization in [9] of property (T) in terms of the geometry of the simplex of invariant measures for associated generalized Bernoulli actions. The emergence of this Poulsen simplex suggests that, if a representation theorem for these processes is available at all, it must take a very different form from the case of set-indexed exchangeable families.