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 $X$.