Wasserstein spaces have the Radon-Nikodym property

Abstract

Wasserstein spaces carry two natural geometries: a vertical one, inherited from the linear structure of signed measures, and a horizontal one, induced by couplings. We prove that a Radon-Nikodym property holds in both. In the vertical geometry, bounded subsets of $P_p(R^d)$ are dentable in an ambient Banach space which itself fails the Radon-Nikodym property; we derive variational principles, Radon-Nikodym theorems for vector measures and martingale convergence results. In the horizontal geometry, we show that a Radon-Nikodym property is inherited from the one of spaces of random variables. We introduce horizontal martingales and prove that they are exactly the laws of $L^p$ -valued martingales.