Some new results on integration for multifunction

TL;DR

This paper extends integration theory for multifunctions by showing that under certain conditions, such multifunctions can be decomposed into strongly measurable selections and integrable multifunctions with specific properties.

Contribution

It introduces new decomposition results for Bochner measurable multifunctions with absolutely continuous variational measure, linking Henstock, Birkhoff, and Pettis integrability.

Findings

Multifunctions with these properties can be expressed as sums of specific integrable components.

The paper establishes connections between different types of integrability for multifunctions.

New conditions for the decomposition of multifunctions into integrable parts are provided.

Abstract

It has been proven in previous papers that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable.