Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions

The aim of this paper is to study relationships among “gauge integrals” (Henstock, McShane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose, we prove the existence of variational...

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Bibliographic Details
Published in:Annali di matematica pura ed applicata 2018-02, Vol.197 (1), p.171-183
Main Authors: Candeloro, D., Di Piazza, L., Musiał, K., Sambucini, A. R.
Format: Article
Language:English
Subjects:
Banach spaces
Decomposition
Integrals
Mathematics
Mathematics and Statistics
Theorems
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Summary:The aim of this paper is to study relationships among “gauge integrals” (Henstock, McShane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose, we prove the existence of variationally Henstock integrable selections for variationally Henstock integrable multifunctions. Using this and other known results concerning the existence of selections integrable in the same sense as the corresponding multifunctions, we obtain three decomposition theorems (Theorems 3.2 , 4.2 , 5.3 ). As applications of such decompositions, we deduce characterizations of Henstock (Theorem 3.3 ) and H (Theorem 4.3 ) integrable multifunctions, together with an extension of a well-known theorem of Fremlin [ 22 , Theorem 8].
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-017-0674-z