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

Abstract

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. As applications of such decompositions, we deduce characterizations of Henstock and $\mathcal{H}$ integrable multifunctions, together with an extension of a well-known theorem of Fremlin.

arXiv comment13 pages

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• A vector Girsanov result and its applications to conditional measures via the Birkhoff integrability
• A Girsanov Result through Birkhoff Integral
• Some remarks on smooth mappings of Hilbert and Banach spaces and their local convexity property