1. Show that if S is the real line R with usual metric, then Theorem 11.7.2 holds for the probability space (0, 1) with Lebesgue measure. Hint: Let Xn be the inverse of the distribution function of Pn as defined in Proposi- tion 9.1.2.
2. Do the same if S is any complete separable metric space. Hint: A countable product of complete separable metric spaces Sn , with product topology, is also metrizable as a complete separable metric space by Theorem 2.5.7.