Journal of Convex Analysis 18 (2011), No. 3, 769--810
Copyright Heldermann Verlag 2011
Pettis Integrability of Multifunctions with Values in Arbitrary Banach Spaces
Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
There is a rich literature describing integrability of multifunctions that take weakly compact convex subsets of a separable Banach space as their values. Most of the papers concern the Bochner type integration, but there is also quite a number of papers dealing with the Pettis integral. On the other hand almost nothing is known in case of non-separable Banach spaces. Only recently the papers of C. Cascales, V. Kadets and J. Rodriguez ["Measurable selectors and set-valued Pettis integral in non-separable Banach spaces", J. Functional Analysis 256 (2009) 673--699; "Measurability and selections of multifunctions in Banach spaces", J. Convex Analysis 17 (2010) 229--240] have been published, where the authors proved the existence of scalarly measurable selections of scalarly measurable multifunctions with weakly compact values. The aim of this paper to fill in partially that gap by presenting a number of theorems that characterize Pettis integrable multifunctions with (weakly) compact non-separable sets as their values. Having applied the above results, I obtained a few convergence theorems, that generalize results known in case of Pettis integrable functions and in case of separably valued multifunctions.
Keywords: Multifunctions, multimeasures, set-valued Pettis integral, support functions, selections.
MSC: 28B20; 28B05, 46G10, 54C60
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