A Choquet theory of Lipschitz‐free spaces

Abstract Let be a complete metric space and let denote the Lipschitz‐free space over . We develop a ‘Choquet theory of Lipschitz‐free spaces’ that draws from the classical Choquet theory and the De Leeuw representation of elements of (and its bi‐dual) by positive Radon measures on , where is the space of pairs , . We define a quasi‐order on the positive Radon measures on that is analogous to the classical Choquet order. Rather than in the classical case where the focus lies on maximal measures, we study the ‐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.