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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some open mapping theorems for measures

Authors: Seymour Ditor and Larry Q. Eifler
Journal: Trans. Amer. Math. Soc. 164 (1972), 287-293
MSC: Primary 46E27; Secondary 28A40
MathSciNet review: 0477729
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Abstract: Given a compact Hausdorff space X, let $ C(X)$ be the Banach space of continuous real valued functions on X with sup norm and let $ M(X)$ be its dual considered as finite regular Borel measures on X. Let $ U(X)$ denote the closed unit ball of $ M(X)$ and let $ P(X)$ denote the nonnegative measures in $ M(X)$ of norm 1. A continuous map $ \varphi $ of X onto another compact Hausdorff space Y induces a natural linear transformation $ \pi $ of $ M(X)$ onto $ M(Y)$ defined by setting $ \pi (\mu )(g) = \mu (g \circ \varphi )$ for $ \mu \in M(X)$ and $ g \in C(Y)$. It is shown that $ \pi $ is norm open on $ U(X)$ and on $ A \cdot P(X)$ for any subset A of the real numbers. If $ \varphi $ is open, then $ \pi $ is $ \mathrm{weak}^*$ open on $ A \cdot P(X)$. Several examples are given which show that generalization in certain directions is not possible. The paper concludes with some remarks about continuous selections.

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Keywords: Open mappings, measures on compact spaces, probability measures, continuous selections
Article copyright: © Copyright 1972 American Mathematical Society

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