Kluv\'{a}nek-Lewis-Henstock integral in a Banach space

TL;DR

This paper explores properties and convergence theorems of the Kluve1nek-Lewis-Henstock integral in Banach spaces, extending it to set-valued measures and establishing new convergence results.

Contribution

It introduces new convergence theorems for the Kluve1nek-Lewis-Henstock integral, including set-valued measure integration and a dominated convergence theorem in Banach spaces.

Findings

Established a.e. convergence version of Dominated and Bounded Convergence Theorems.

Extended the integral to scalar-valued functions with set-valued measures.

Proved a - dominated convergence theorem for set-valued integrals.

Abstract

We investigate some properties and convergence theorem of Kluv\'{a}nek-Lewis-Henstock $\m-$integrability for $\m-$measurable functions that we introduced in \cite{ABH}. We give a $\m-$a.e. convergence version of Dominated (resp. Bounded) Convergence Theorem for $\m.$ We introduce Kluv\'{a}nek-Lewis-Henstock integrable of scalar-valued functions with respect to a set valued measure in a Banach space. Finally we introduce $(KL)-$type Dominated Convergence Theorem for the set-valued Kluv\'{a}nek-Lewis-Henstock integral.