Let $X$ be a complete separable metric space. Given two sequence of Borel probability measures $\{\mu_{1,n}\}$ and $\{\mu_{2,n}\}$ converge to $\mu_1$ and $\mu_2$ in pointwise topology, that is, for each $i=1,2$, $$\mu_{i,n}(A) \rightarrow \mu_{i}(A)$$ for every measurable subset $A$ of $X$.
Does the product measure $\mu_{1,n} \times \mu_{2,n}$ converges to $\mu_1 \times \mu_2 $ in pointwise topology? i.e. for every measurable subset $E$ of $X \times X$, $$(\mu_{1,n} \times \mu_{2,n})(E) \rightarrow (\mu_1 \times \mu_2)(E)$$