Lifting the functors \(U_\tau\) and \(U_R\) to the categories of bounded metric spaces and uniform spaces (Q2736060)

The author studies the problem of metrizability of the functor \(U_\tau\) assigning to each Tychonov space \(X\) the space \(U_\tau(X)\) of all \(\tau\)-additive measures \(\mu\) on \(X\) with \(\mu(X)\leq 1\). The categorial properties of this functor and its subfunctor \(U_R\) of Radon measures were studied by the author in [Vestnik Moskov Univ. 3, 38-42 (1999; Zbl 0949.54043)] and [Topology Appl. 107, 131-145 (2000; Zbl 0976.46015)]. In this paper he defines a canonical formula extending each bounded continuous pseudometric \(\rho\) on \(X\) to a bounded continuous pseudometric \(U_\tau(\rho)\) on \(U_\tau(X)\) and shows that for a bounded (complete) metric \(\rho\) generating the topology of \(X\) the metric \(U_\tau(\rho)\) generates the topology of the space \(U_\tau(X)\) (and is complete). Restricted to the space \(P_\tau(X)\) of probability \(\tau\)-additive measures on \(X\) the metric \(U_\tau(\rho)\) coincides with the classical Kantorovich metric. Using his extension formula the author constructs liftings of the functors \(U_\tau\) and \(U_R\) to the category of bounded metric spaces and their (uniformly) continuous maps and also to the category of uniform spaces and their uniformly continuous maps. Analogous liftings of the functor \(P_\tau\) of probability \(\tau\)-additive measures were constructed by the reviewer in [Mat. Stud. 5, 88-106 (1995; Zbl 1023.28502)]. At the end of the paper the author remarks that like the functor \(P_\tau\) (see \textit{V. Fedorchuk} [Topology Appl. 91, 25-45 (1999; Zbl 0986.54032)]), the functor \(U_\tau\) preserves the completeness of metric spaces but fails to preserve the completeness of uniform spaces.