Kantorovich duality for optimal transport on completely regular Hausdorff spaces

Abstract: We introduce a new intermediate optimization problem situated between Kantorovich's primal and dual formulations. This new problem extends Kantorovich's duality to separable Baire measures, which are strictly more general than tight (or Radon) measures in completely regular Hausdorff spaces. In the special case where the measures are Radon, our intermediate problem aligns with the classical Kantorovich's primal problem. Existence of solutions for all three formulations are also provided within this comprehensive framework.