Entropy under Disintegrations

Juan Pablo Vigneaux

Abstract. We consider the differential entropy of probability measures absolutely continuous with respect to a given σ-finite “reference” measure on an arbitrary measure space. We state the asymptotic equipartition property in this general case; the result is part of the folklore but our presentation is to some extent novel. Then we study a general framework under which such entropies satisfy a chain rule: disintegrations of measures. We give an asymptotic interpretation for conditional entropies in this case. Finally, we apply our result to Haar measures in canonical relation.