Def.A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set $U\subseteq X$, $\mu(U)$ = sup$\{\mu(A): A\in\mathcal{A}$ and $A\subseteq U\}$.

My question is: To what extent the following result will be true?

Q:Let $\{X_\alpha:a\in A\}$ be a family of compact Hausdorff space with uniformly regular Radon measure $\mu_\alpha$. The product $X=\prod_{\alpha\in A}X_\alpha$ is uniformly regular.

I think **Q** is true when $A$ is countable, but I need a sketch of the prove. The most important case when $A$ is uncountable.

Please share your ideas.