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

TL;DR

This paper investigates the geometric structure of cube-exchangeable measures, revealing that unlike other exchangeability classes, their set of extreme points is dense, indicating a fundamentally different representation theory.

Contribution

It proves that cube-exchangeability measures form a Poulsen simplex, contrasting with the Bauer simplex structure in other exchangeability contexts, highlighting the need for new representation approaches.

Findings

Cube-exchangeable measures form a Poulsen simplex.

Extreme points of these measures are dense.

Standard representation theorems do not directly apply.

Abstract

In his 1985 survey 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, arrays and set-indexed families. 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 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.