On some categorical properties of uniform spaces of probability measures (Q1379784)

For a Tychonoff space \(X\), denote by \(P_\beta(X)\) the set of all probability measures on \(X\) with compact support. Let \(P_\beta(X)\) be equipped with the weak star topology. If \(Y\) is another Tychonoff space and \(f:X\to Y\) a continuous map, then \(P_\beta(f)\) is defined by \(P_\beta(f)(\mu)= \mu\circ f^{-1}\). \(P_\beta\) is a covariant functor acting in the category Tych of Tychonoff spaces. This functor can be lifted to a functor \(P_\beta^u\) onto the category Unif of uniform spaces, i.e. \(F\circ P_\beta^u= P_\beta\circ F\) where \(F\) denotes the ``forgetful'' functor. One of the main results says that there is no ``embedding'' functor \(U:\text{Tych}\to \text{Unif}\) such that \(P_\beta^u\circ U= U\circ P_\beta\). Another result gives a characterization of precompactness of a uniform space involving the functor \(P_\beta^u\).