Functors of probability measures in topological categories
The work represents a detailed survey of topological properties of probability measure spaces and their maps. It is organized as follows. Section 1 deals with preliminaries and the proof of normality of the functor \(P\) of probability measures in the category \(Comp\) of compacts. Section 2 is devoted to general properties of the functors \(P_\beta , P_r, P_\tau,\) where \(P_\beta (X)\) is the space of probability measures on \(X\) with compact supports, \(P_r\)(X) is the space of Radon probability measures on \(X\) , while \(P_\tau (X)\) is the space of \(\tau\)-smooth probability measures on \(X\). In Section 3 the author gives constructions permitting to lift the functor \(P_\beta\) to the categories \(Metr_b\) of bounded metric spaces and \(Metr_{bu}\) of bounded metric spaces and uniformly continuous mappings. In Section 4 results of Section 3 are used to lift the functor \(P_\beta\) to the category \(Unif\) of uniform spaces. Section 5 contains many results with detailed proofs. Among them is the author's theorem characterizing Tychonoff bundles in the class of mappings of the form \(P(f)\). Section 6 deals with the mapping of barycenter. Section 7 contains a survey of new results.
scientific article; zbMATH DE number 1231430
scientific article