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

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...

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Veröffentlicht in:Topology and its applications 2021-05, Vol.295, p.107673, Article 107673
1. Verfasser: Goubault-Larrecq, Jean
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Sprache:eng
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Continuous valuation
General Topology
Kantorovich-Rubinstein quasi-metric
Mathematics
Probability
Quasi-metric
Weak topology
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description 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.
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subjects Continuous valuation
General Topology
Kantorovich-Rubinstein quasi-metric
Mathematics
Probability
Quasi-metric
Weak topology
title Kantorovich-Rubinstein quasi-metrics I: Spaces of measures and of continuous valuations
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