The topology of spaces of probability measures, I

For a Tychonoff space $X$, the constructions $\widehat{P}(X)$ and $P_{\tau }(X)$ of the spaces of probability Radon measures and probability $\tau$-smooth measures on $X$ are considered. It is proved that these constructions determine functors in the category of Tychonoff spaces, which extend the functor $P$ of probability measures in the category of compacta. In this part we investigate general topological properties of the spaces $\widehat{P}(X)$ and $P_{\tau }(X)$, as well as categorial properties of the functors $\widehat{P}$ and $P_{\tau }$.