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Transactions of the American Mathematical Society

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Representation of set valued operators

Author: Nikolaos S. Papageorgiou
Journal: Trans. Amer. Math. Soc. 292 (1985), 557-572
MSC: Primary 47H99; Secondary 28B20, 46E30, 46G99
MathSciNet review: 808737
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Abstract: In this paper we prove representation theorems for set valued additive operators acting on the spaces $ L_X^1(X = {\text{separable Banach space)}}$, $ {L^1}$ and $ {L^\infty }$. Those results generalize well-known ones for single valued operators and among them the celebrated Dunford-Pettis theorem. The properties of these representing integrals are studied. We also have a differentiability result for multifunctions analogous to the one that says that an absolutely continuous function from a closed interval into a Banach space with the Radon-Nikodým property is almost everywhere differentiable and also it is the primitive of its strong derivative. Finally we have a necessary and sufficient condition for the set of integrable selectors of a multifunction to be $ w$-compact in $ L_X^1$. This result is a new very general result about $ w$-compactness in the Lebesgue-Bochner space $ L_X^1$.

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Keywords: Set valued operator, lifting, support function, measurable multifunction integrable selector, Radon-Nikodým property, weak compactness, multimeasure
Article copyright: © Copyright 1985 American Mathematical Society