2024/12/06 by R. Jeffrey Smith, Smith, Richard J.
Mathematics · Computer Science · #Advanced Topology and Set Theory #Advanced Banach Space Theory #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2412.05177
Let (M,d) be a complete metric space and let 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 F(M) (and its bidual) by positive Radon measures on β\widetildeM, where \widetildeM is the space of pairs (x,y) ∈ M × M, x ≠ y. We define a quasi-order \preccurlyeq on the positive Radon measures on β\widetildeM 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.