Open mapping theorems for probability measures on metric spaces
Larry Q. Eifler · Pacific Journal of Mathematics · 1976
Let 5 and T denote complete separable metric spaces.Let P(S) denote the collection of probability measures on S and equip P(S) with the weak topology.It φ: S -» T is continuous and onto, then φ induces a weakly continuous mapping φ° of P(S) onto P(T).We show that φ° is open in the weak topology if and only if φ is open.However, φ° is always open in the norm topology.Let K be a totally disconnected compact metric space and let S κ denote the set of continuous mappings of K into S. Then there exists a natural mapping π of P(S K ) into P(S) K .Blumenthal and Corson have shown that π is onto.We establish that π is an open mapping in the weak topology.