Topology of probability measure spaces, II

This paper is a follow-up to the author's work "Topology of probability measure space, I" devoted to investigation of the functors $\hat P$ and $P_\tau$ of spaces of probability $\tau$-smooth and Radon measures. In this part, we study the barycenter map for spaces of Radon probability measures. The obtained results are applied to show that the functor $\hat P$ is monadic in the category of metrizable spaces. Also we show that the functors $\hat P$ and $P_\tau$ admit liftings to the category $BMetr$ of bounded metric spaces and also to the category $Unif$ of uniform spaces, and investigate properties of those liftings.