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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On exposed points of the range of a vector measure. II

Author: R. Anantharaman
Journal: Proc. Amer. Math. Soc. 55 (1976), 334-338
MathSciNet review: 0399851
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Abstract: If a weakly compact convex set $ K$ in a real Banach space $ X$ is strongly exposed by a dense set of functionals in $ X'$, it is proved that the functionals which expose $ K$ form a residual set in $ X'$. If $ v:\mathcal{A} \to X$ is a measure, it follows that the set of exposing functionals of its range is a residual $ {G_\delta }$ in $ X'$. This, in turn, is found to be equivalent to a theorem of B. Walsh on the residuality of functionals $ x' \in X'$ for which $ x' \circ v \equiv v$.

If the set of exposed points of $ v(\mathcal{A})$ is weakly closed and $ {v_A}$ is the restriction of $ v$ to any set $ A \in \mathcal{A}$, it is further proved that every exposed point of the range of $ {v_A}$ is of the form $ v(A \cap E)$, where $ E \in \mathcal{A}$ and $ v(E)$ is an exposed point of $ v(\mathcal{A})$.

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Keywords: Exposed points, exposing functionals, range of vector measures, weakly compact convex sets
Article copyright: © Copyright 1976 American Mathematical Society