Compactification of Spaces of Measures and Pseudocompactness

We prove pseudocompactness of a Tychonoff space X and the space \(\mathcal{P}(X)\) of Radon probability measures on it with the weak topology under the condition that the Stone–Čech compactification of the space \(\mathcal{P}(X)\) is homeomorphic to the space \(\mathcal{P}(\beta X)\) of Radon probability measures on the Stone–Čech compactification of the space X.