Integration of multifunctions with closed convex values in arbitrary Banach spaces

TL;DR

This paper explores the integral properties of multifunctions with closed convex values in Banach spaces, extending existing techniques and providing new decomposition results and integrability characterizations.

Contribution

It introduces new methods for analyzing multifunctions in Banach spaces, especially for positive and vector-determined multifunctions, and characterizes McShane integrability via Henstock and Pettis integrals.

Findings

Decomposition results for scalar and gauge integrals of multifunctions

Full characterization of McShane integrability in terms of Henstock and Pettis integrals

Extension of integral analysis techniques to more general Banach space multifunctions

Abstract

Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given.