Stone-Cech Compactifications of Spaces of Probability Measures

It is proved that $P(\beta X)=\beta P(X)$ if and only if $P(X)$ is a pseudocompact space, where $P(X)$ is the space of Radon probability measures with the weak topology and $\beta X$ is the Stone--\v Cech compactification of $X$.

A locally compact pseudocompact space $X$ is constructed such that $P(X)$ is not pseudocompact.

Conditions are obtained under which $P(X)$ is pseudocompact.