On compactification of spaces of measures

In this paper, we compare the Stone–Čech compactification $\beta \mathcal{P}(X)$ of the space $\mathcal{P}(X)$ of Radon probability measures on a Tychonoff space $X$, equipped with the weak topology, with the space $\mathcal{P}(\beta X)$ of Radon probability measures on the Stone–Čech compactification $\beta X$ of the space $X$. It is shown that for any noncompact metric space $X$, the compactification $\beta \mathcal{P}(X)$ does not coincide with $\mathcal{P}(\beta X)$. We discuss the case of more general Tychonoff spaces and also the case of the Samuel compactification, for which the coincidence holds.