Vol. 66, No. 1, 1976

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Open mapping theorems for probability measures on metric spaces

Larry Quin Eifler

Vol. 66 (1976), No. 1, 89–97

Let S 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. If φ : S T is continuous and onto, then φ induces a weakly continuous mapping φ0 of P(S) onto P(T). We show that φ0 is open in the weak topology if and only if φ is open. However, φ0 is always open in the norm topology. Let K be a totally disconnected compact metric space and let SK denote the set of continuous mappings of K into S. Then there exists a natural mapping π of P(SK) 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.

Mathematical Subject Classification 2000
Primary: 28A32, 28A32
Secondary: 46E30
Received: 27 January 1976
Published: 1 September 1976
Larry Quin Eifler