Kantorovich-Rubinstein quasi-metrics I: Spaces of measures and of continuous valuations

We show that the space of subprobability measures, equivalently of subprobability continuous valuations, on an algebraic (resp., continuous) complete quasi-metric space is again algebraic (resp., continuous) and complete, when equipped with the Kantorovich-Rubinstein quasi-metrics dKR (unbounded) or dKRa (bounded), themselves asymmetric forms of the well-known Kantorovich-Rubinstein metric. We also show that the dKR-Scott and the dKRa-Scott topologies then coincide with the weak topology. We obtain similar results for spaces of probability measures, equivalently of probability continuous valuations, with the dKRa quasi-metrics, or with the dKR quasi-metric under an additional rootedness assumption.