Lecture 21: Tightness of measures

Given two metric spaces S1, S2 and a measurable function f : S1 → S2, sup­ pose S1 is equipped with some probability measure P. This induces a proba­ bility measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A)) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf−1 [X] is equal to EP[X(f)]. (Convince your­ self that this is the case by looking at the special case when f is a simple func­ tion). Theorem 1 (Mapping Theorem). Suppose Pn ⇒ P for a sequence of probability measures P, Pn on S1 and suppose f : S1 → S2 is continuous. Then Pnf ⇒ Pf−1 on S2. Proof. We use Portmentau theorem, in particular weak convergence character­ ization using bounded continuous functions. Thus let g : S2 → R be any bounded continuous function. Since it is continuous, it is also measurable, thus it is also a random variable defined on (S2, B2), where B2 is the Borel σ-field on S2. We have, EPnf−1 [g] = EPn [g(f)]. Since g is a bounded continuous, then the composition is also bounded continu­ ous. Therefore, by Portmanteau theorem