Permutation invariant boolean states

Abstract

We give a self-contained proof of the following result: Finitely additive probability measures (also known as “states”) of the free boolean algebra \({\mathsf F}_\omega \) over the free generating set \(\{X_1,X_2,\ldots \}\) having the invariance property under finite permutations of the \(X_i\), coincide with states lying in the closure of the set of convex combinations of product states of \({\mathsf F}_\omega \) in the vector space \(\mathbb R^{{\mathsf F}_\omega }\) equipped with the product topology. De Finetti’s celebrated exchangeability theorem can be easily recovered from our proof.