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Set valued measures and integral representation. (English) Zbl 0885.28008
The present paper is based on an earlier paper of F. Hiai [J. Multivariate Anal. 8, 96-118 (1978; Zbl 0384.28006)]. Let $$X$$ be a Banach space. $$\Sigma$$ denotes a $$\sigma$$-algebra of subsets of a non-empty set $$\Omega$$, $${\mathcal P}_a(X)$$ the family of all non-empty subsets of $$X$$ and $${\mathcal P}_{wee}(X)$$ that of all weakly compact convex subsets of $$X$$. The authors obtain a sufficient condition to ensure the unconditional convergence of $$\sum^\infty_{n= 1}A_n= \{\sum x_n: x_n\in A_n\}$$, where $$\{A_n\}^\infty_1$$ is a uniformly bounded family of subsets of $$X$$ and deduce that a boundedly $$\sigma$$-additive set-valued measure $$M:\Sigma\to {\mathcal P}_{wee}(X)$$ is strongly additive and a bounded mapping $$M:\Sigma\to{\mathcal P}_{wee}(X)$$ is $$\sigma$$-additive if and only if $$M$$ is weakly countably additive. When $$X$$ has RNP and $$M:\Sigma\to{\mathcal P}_a(X)$$ is $$\sigma$$-additive, non-atomic and of bounded variation, then they show that $$\text{cl }M(E)$$ is convex for all $$E\in \Sigma$$, whose proof is based on Theorem 1.2 of F. Hiai (op. cit.). They also prove a result similar to Proposition 2.1 of F. Hiai (op. cit.) when $$M:\Sigma\to{\mathcal P}_{wee}(X)$$ is bounded and $$\sigma$$-additive (in Lemma 2.1 the hypothesis of boundedness should be included for its validity) and then obtain the selection theorem for $$M$$. Using these results, the authors generalize the classical extension theorem known for $$X$$-valued vector measures defined on an algebra of sets to set-valued measures.
Introducing the concept of a $$\mu$$-weakly compactly separable bounded set-valued measure $$M:\Sigma\to{\mathcal P}_{wee}(X)$$, where $$\mu$$ is a finite positive measure on $$\Sigma$$, they show that whenever $$X$$ has separable dual and $$M:\Sigma\to{\mathcal P}_{wee}(X)$$ is a $$\mu$$-continuous set-valued measure, there exists a measurable weakly integrable bounded set-valued function $$F:\Omega\to{\mathcal P}_{wee}(X)$$ such that $M(A)= (W)\int_A Fd\mu$ if and only if $$M$$ is $$\mu$$-weakly compactly separable, where $$(W)\int_A Fd\mu$$ is a Pettis-Aumann type integral.

##### MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 46G10 Vector-valued measures and integration 28B05 Vector-valued set functions, measures and integrals
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