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.
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Acknowledgments
I am grateful to K. A. Afonin and S. N. Popova for useful discussions.
Funding
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/.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2024, Vol. 58, pp. 4–21 https://doi.org/10.4213/faa4184.
Dedicated to Anatoly Moiseevich Vershik, on his 90th birthday
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Bogachev, V. On Compactification of Spaces of Measures. Funct Anal Its Appl 58, 2–15 (2024). https://doi.org/10.1134/S0016266324010027
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DOI: https://doi.org/10.1134/S0016266324010027