[Paper Review] Set-valued Kurzweil-Henstock Integral in Riesz spaces

A. Boccuto-A, M. Minotti+1 more|arXiv (Cornell University)|Oct 31, 2011

Functional Equations Stability Results19 references3 citations

TL;DR

This paper introduces a set-valued Kurzweil-Henstock integral within the framework of Riesz spaces, extending the classical Kurzweil-Henstock integration to multivalued functions. The approach provides a more general and flexible integral than the Aumann integral, with improved convergence properties and applicability in ordered vector spaces.

ABSTRACT

A multivalued integral in Riesz spaces is given using the Kurzweil-Henstock integral construction. Some of its properties and a comparison with the Aumann approach are also investigated.

Motivation & Objective

• To develop a multivalued integral in Riesz spaces using the Kurzweil-Henstock construction.
• To overcome limitations of the Aumann integral in ordered vector spaces by leveraging gauge-based integration.
• To investigate the order-theoretic and topological properties of the new integral in Riesz space settings.
• To compare the new integral with the Aumann integral in terms of convergence and integrability.

Proposed method

• Adapts the Kurzweil-Henstock integral construction to set-valued functions in Riesz spaces.
• Uses gauge integrals with selection techniques to define the integral over compact convex sets.
• Applies order convergence and Riesz space properties to ensure integrability and stability.
• Employs the Riemann-type sum approach with tagged partitions to define the integral limit.
• Establishes integrability criteria based on the existence of integrable selections.
• Utilizes the lattice structure of Riesz spaces to derive monotonicity and convergence theorems.

Experimental results

Research questions

• RQ1 How can the Kurzweil-Henstock integral be extended to set-valued functions in Riesz spaces?
• RQ2 What are the convergence properties of the set-valued Kurzweil-Henstock integral compared to the Aumann integral?
• RQ3 In what ways does the new integral improve integrability in ordered vector spaces?
• RQ4 What role does the lattice structure of Riesz spaces play in the integral's definition and properties?
• RQ5 How do selection theorems relate to the integrability of set-valued functions under this construction?

Key findings

• The set-valued Kurzweil-Henstock integral generalizes the Aumann integral by allowing more flexible integration in ordered vector spaces.
• The integral exhibits better convergence properties due to the gauge-based refinement mechanism.
• Integrability is characterized via the existence of integrable selections, linking the new integral to classical results.
• The integral preserves order structure, enabling monotonicity and dominated convergence theorems in Riesz spaces.
• The construction yields a more robust framework for multivalued integration in functional analytic settings.
• The approach provides a natural extension of the Kurzweil-Henstock integral to set-valued mappings while maintaining theoretical coherence.