Probability measures on product spaces with uniform metrics

For a countable product of complete separable metric spaces with a topology induced by a uniform metric, the set of Borel probability measures coincides with the set of completions of probability measures on the product o-algebra. Whereas the product space with the uniform metric is non-separable, the support of any Bofrel measure is separable, and the topology of weak convergence on the space of Borel measures is metrizable by both the Prohorov metric and the bounded Lipschitz metric.