  For a Tychonoff space $X$, the constructions $\hat P(X)$ 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 $\hat P(X)$ and $P_\tau(X)$, as
well as categorial properties of the functors $\hat P$ and $P_\tau$.
