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Henstock-Kurzweil-Pettis integrability of compact valued multifunctions with values in an arbitrary Banach space. (English) Zbl 1309.28012

Summary: The aim of this paper is to describe Henstock-Kurzweil-Pettis (HKP) integrable compact valued multifunctions. Such characterizations are known in case of functions (see the authors [Stud. Math. 176, No. 2, 159–176 (2006; Zbl 1118.26008)]). It is also known (see the authors [in: G. P. Curbera (ed.) et al., Vector measures, integration and related topics. Selected papers from the 3rd conference on vector measures and integration. Basel: Birkhäuser. Operator Theory: Advances and Applications 201, 171–182 (2010; Zbl 1248.28019)]) that each HKP-integrable compact valued multifunction can be represented as a sum of a Pettis integrable multifunction and of an HKP-integrable function. Invoking to that decomposition, we present a pure topological characterization of integrability. Having applied the above results, we obtain two convergence theorems, that generalize results known for HKP-integrable functions. We emphasize also the special role played in the theory by weakly sequentially complete Banach spaces and by spaces possessing the Schur property.

MSC:

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
46G10 Vector-valued measures and integration
47H04 Set-valued operators
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