We prove pseudocompactness of a Tychonoff space X and the space P(X) of Radon probability measures on it with the weak topology under the condition that the Stone-{C}ech compactification of the space P(X) is homeomorphic to the space P(β X) of Radon probability measures on the Stone-{C}ech compactification of the space~X.
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В. И. Богачев (2024) studied this question.
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