A Choquet theory of Lipschitz-free spaces

Abstract

Let $(M,d)$ be a complete metric space and let $\mathcal{F}(M)$ denote the Lipschitz-free space over $M$. We develop a ``Choquet theory of Lipschitz-free spaces'' that draws from the classical Choquet theory and the De Leeuw representation of elements of $\mathcal{F}(M)$ (and its bidual) by positive Radon measures on $\beta\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y) \in M \times M$, $x \neq y$. We define a quasi-order $\preccurlyeq$ on the positive Radon measures on $\beta\widetilde{M}$ that is analogous to the classical Choquet order. Rather than in the classical case where the focus lies on maximal measures, we study the $\preccurlyeq$-minimal measures and show that they have a host of desirable properties. Among the applications of this theory is a solution (given elsewhere) to the extreme point problem for Lipschitz-free spaces.