Measure-theoretic Lie brackets for nonsmooth vector fields

In this paper we prove a generalization of the classical notion of commutators of vector fields in the framework of measure theory, providing an extension of the set-valued Lie bracket introduced by Rampazzo-Sussmann for Lipschitz continuous vector fields. The study is motivated by some applications to control problems in the space of probability measures, modeling situations where the knowledge of the state is probabilistic, or in the framework of multi-agent systems, for which only a statistical description is available. Tools of optimal transportation theory are used.