Abstract.

Abstract. It is shown that the obvious generalization of the Pettis integral of a multifunction obtained by replacing the Lebesgue integrability of the support functions by the Kurzweil-Henstock integrability, produces an integral which can be described -in case of multifunctions with (weakly) compact convex values -in terms of the Pettis set-valued integral. Mathematics Subject Classifications (2000):Primary: 28B20; secondary: 26A39, 28B05, 46G10, 54C60.

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The aim of this paper is to study relationships among "gauge integrals" (Henstock, Mc Shane, 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 (Theorem 3.2, Theorem 4.2 and Theorem 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].