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

Abstract 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 d KR (unbounded) or d KR a (bounded), themselves asymmetric forms of the well-known Kantorovich-Rubinstein metric. We also show that the d KR -Scott and the d KR a -Scott topologies then coincide with the weak topology. We obtain similar results for spaces of probability measures, equivalently of probability continuous valuations, with the d KR a quasi-metrics, or with the d KR quasi-metric under an additional rootedness assumption.

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