Spaces of Probability Measures

Abstract

Let S be a Polish space with a complete metric d taking values in [0,1] and P(S) the space of probability measures on S. Recall the map h : S → [0,1] ∞ of Theorem 1.1.1. Since \( \overline {h(S)} \) is compact, \( \overline {C(h(s)} ) \) is separable. Let fi be countable dense in the unit ball of \( \overline {C(h(s)} ) \) and {f′ i} their restrictions to h(h). Define {fi} ⊂ Cb(S) (= the space of bounded continuous functions S → R) by fi = f′i o h, i ≥ 1.