A note on set-valued Henstock--McShane integral in Banach (lattice) space setting

TL;DR

This paper investigates set-valued Henstock--McShane integrals in Banach lattice spaces, providing new proofs, comparison with Aumann integrals, and existence results for functions on [0,1].

Contribution

It introduces a R{ a}dstr"{o}m-type embedding theorem to simplify multivalued integrals and extends the theory to order-type integrals in Banach lattices.

Findings

Established a R{ a}dstr"{o}m-type embedding theorem.

Provided new proofs for decomposition results.

Obtained existence results for functions on [0,1].

Abstract

We study Henstock-type integrals for functions defined in a Radon measure space and taking values in a Banach lattice $X$. Both the single-valued case and the multivalued one are considered (in the last case mainly $cwk(X)$-valued mappings are discussed). The main tool to handle the multivalued case is a R{\aa}dstr\"{o}m-type embedding theorem established in [50]: in this way we reduce the norm-integral to that of a single-valued function taking values in an $M$-space and we easily obtain new proofs for some decomposition results recently stated in [33,36], based on the existence of integrable selections. Also the order-type integral has been studied: for the single-valued case some basic results from [21] have been recalled, enlightning the differences with the norm-type integral, specially in the case of $L$-space-valued functions; as to multivalued mappings, a previous definition ([6]) is restated in an equivalent way, some selection theorems are obtained, a comparison with the Aumann integral is given, and decompositions of the previous type are deduced also in this setting. Finally, some existence results are also obtained, for functions defined in the real interval $[0,1]$.