Distances of Probability Measures and Random Variables

Let (S, d) be a separable metric space. Let \( P; > \left( S \right)\) be the set of Borel probability measures on S. \(C\left( S \right)\) denotes the Banach space of bounded continuous real-valued functions on S, with norm $$\left\| f \right\|_\infty= \sup \left\{ {\left| {f\left( x \right)} \right|:x{\text{ }}\varepsilon {\text{ }}S} \right\}.$$