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~$X$.