Bernoulli shifts with bases of equal entropy are isomorphic

Abstract

We prove that if $G$ is a countably infinite group and $(L, \lambda)$ and $(K, \kappa)$ are probability spaces having equal Shannon entropy, then the Bernoulli shifts $G \curvearrowright (L^G, \lambda^G)$ and $G \curvearrowright (K^G, \kappa^G)$ are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both $\lambda$ and $\kappa$ have at least $3$ points in their support. We furthermore produce finitary isomorphisms in the case where both $L$ and $K$ are finite.