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

我们证明，当配备了Kantorovich-Rubinstein准方程时，在代数（分别为连续的）完全拟度量空间上，子概率测度的空间（等同于子概率连续计价）又是代数的（分别为连续的）且是完备的。指标 $d_{\text{KR}}$ （无界）或 $d_{\text{KR}}^{\mathrm{一个}}$（有界），它们本身是众所周知的Kantorovich-Rubinstein度量的不对称形式。我们还表明$d_{\text{KR}}$-斯科特和 $d_{\text{KR}}^{\mathrm{一个}}$然后，Scott拓扑与弱拓扑重合。对于概率测度空间，我们获得了相似的结果，等效于概率连续估值，其中$d_{\text{KR}}^{\mathrm{一个}}$ 准度量，或与 $d_{\text{KR}}$ 附加生根假设下的准度量。