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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.

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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