  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.
