Distances on probability measures and random variables (Q607299)

Let \(S\) be a separable metric space and \(P(S)\) the set of all Borel probability measures on \(S\). The authors define a distance function \(d_w\) between points in \(P(S)\) and subsets of \(P(S)\) which they call weak distance. This distance is usually not related with a metric on \(P(S)\) and is related with the usual Prohorov metric \(\rho\) and the usual total variation distance \(d_{TV}\) by \(\rho\leq d_w\leq d_{TV}\). The authors also investigate relative compactness, tightness, Prohorov's theorem and many further aspects for this weak distance. Editorial remark: See also the Publisher's note [ibid. 159, No. 10-11, 2557 (2012; Zbl 1244.60007)].