Uniform spaces of probabilistic measures. (English. Russian original) Zbl 0907.54023

Sadovnichij, Yu. V.

Mosc. Univ. Math. Bull. 50, No. 2, 55-56 (1995); translation from Vestn. Mosk. Univ., Ser. I 1995, No. 2, 85-87 (1995).

For a Tikhonov space \(X\) by \(P(X)\) is denoted the set of all probability measures \(\mu\in P(bX)\) (where \(bX\) is an arbitrary compact extension, e.g. the Stone-Čech compactification) the supports of which lie in \(X\). This set is endowed with a topology which is induced by the weak* topology of the compact space \(P(bX)\). Let \((X,\rho)\) be a metric space with \(\text{diam } X\leq 1\). On \(P(X)\) a metric \(P(\rho)\) is defined 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.
In the paper [Mosc. Univ. Math. Bull. 49, No. 4, 27-29 (1994); translation from Vestn. Mosk. Univ., Ser. I 1994, No. 4, 31-34 (1994; Zbl 0907.54026)] the author proved that \((P(X), P(\rho))\) is a topological space with the weak-* topology.
In this paper he generalizes this theorem to uniform spaces. For a uniform space \((X,U)\) let \(R_U= \{\rho_\alpha:\alpha\in A\}\) be the family of all uniformly continuous bounded pseudometrics. Then the following theorem is valid: The family of pseudometrics \(P(R_U)= \{P(\rho_\alpha): \alpha\in A\}\) generates on \(P(X)\) some uniformity \(P(U)\).

MSC:

54E15	Uniform structures and generalizations
28A33	Spaces of measures, convergence of measures
60B05	Probability measures on topological spaces

Citations:

Zbl 0907.54026