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.