Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures

TL;DR

This paper extends Henstock-type integrals to multivalued functions in Banach lattices with non-atomic measures, utilizing a Rådström-type embedding to simplify the analysis and derive new decomposition results.

Contribution

It introduces a Henstock multivalued integrability framework in Banach lattices using a Rådström embedding, enabling reduction to single-valued integrals and new decomposition proofs.

Findings

Established a Rådström-type embedding theorem for multivalued integrals.

Reduced multivalued integrals to single-valued integrals in an M-space.

Provided new proofs for decomposition results in multivalued integration.

Abstract

Henstock-type integrals are considered, for multifunctions taking values in the family of weakly compact and convex subsets of a Banach lattice $X$. The main tool to handle the multivalued case is a R{\aa}dstr\"om-type embedding theorem established by C. C. A. Labuschagne, A. L. Pinchuck, C. J. van Alten in 2007. In this way the norm and order integrals reduce to that of a single-valued function taking values in an $M$-space, and new proofs are deduced for some decomposition results recently stated in two recent papers by Di Piazza and Musial based on the existence of integrable selections.