Abstract:
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.
Keywords:Radon measure, weak topology, Stone–Čech compactification, Samuel compactification, compactification of the space of measures.
This research is supported by the Russian Science Foundation grant no. 22-11-00015 (at Lomonosov Moscow State University),
https://rscf.ru/en/project/22-11-00015/.