A metric on the space of probability measures (Q1358392)

For \(X\) a Tikhonov space and \(bX\) an arbitrary compactification of \(X\) (for example the Stone-Čech compactification), with \(P(X)\) is denoted the set of all probability measures \(\mu\in P(bX)\), the supports of which are contained in \(X\). \(P(X)\) is endowed with a topology induced by the weak* topology of the compact space \(P(bX)\). Let \((X,\rho)\) be a metric space and \(\text{diam } X\leq 1\). The author defines a metric \(P(\rho)\) on \(P(X)\) by \[ P(\rho) (\mu_1,\mu_2)= \inf\left\{ \int_{X\times X}\rho(x_1,x_2) d\lambda: \lambda\in\Lambda (\mu_1,\mu_2) \right\} \] where \(\Lambda(\mu_1,\mu_2)= \{\lambda\in P(X\times X): \text{pr}_i(\lambda)= \mu_i\), \(i=1,2\}\), here \(\text{pr}_i= P(p_i)\), where \(p_i: X\times X\to X\) is the projection onto the \(i\)-th factor. \textit{Y. Al'Kassas} proved that the space \(P(X)\) is metrizable [Mosc. Univ. Math. Bull. 48, No. 1, 12-24 (1993); translation from Vestn. Mosk. Univ., Ser. I 1993, No. 1, 14-17 (1993; Zbl 0823.54007)]. In this paper, directly a metric generating the topology of this space is determined. Theorem: The topology of the metric space \((P(X), P(\rho))\) is the weak* topology.