Let $S$ be a topological space. Denote by $\Delta(S)$ the set of probability measures on the Borel $\sigma$-algebras of $S$, endowed with its weak$^\star$-topology $\tau_S$.
Question. Let $X,Y$ be compact metric spaces. Is it true that the restriction of the weak$^\star$-topology on $\Delta(X\times Y)$ to $\Delta(X)\times \Delta(Y)$ coincides with the product topology $\tau_X \times \tau_Y$?