Entropy under disintegrations

02/18/2021

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by   Juan Pablo Vigneaux, et al.

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We consider the differential entropy of probability measures absolutely continuous with respect to a given σ-finite reference measure on an arbitrary measurable 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.