[Paper Review] Some Remarks on Vector-Valued Integration

Vladimir Kadets, B. M. Shumyatskiy+3 more|ArXiv.org|Mar 17, 1999

Advanced Control Systems Optimization2 references17 citations

TL;DR

This paper introduces the RL (Riemann-Lebesgue) integral for Banach space-valued functions, defined via upper Lebesgue integrals of the norm, and establishes its properties. The key contribution is proving that RL-integrability implies Pettis integrability, and that RL-integrable functions are weakly measurable, providing a bridge between Riemann, Bochner, and Pettis integration with strong structural implications for Banach space geometry.

ABSTRACT

The article presents a new method of integration of functions with values in Banach spaces. This integral and related notions prove to be a useful tool in the study of Banach space geomtry.

Motivation & Objective

• To develop a new vector-valued integration theory that unifies Riemann and Bochner integrals while avoiding the limitations of Pettis integration.
• To establish a framework based on upper Lebesgue integrals of the norm, leading to a new Banach space of functions—the RL space.
• To prove that RL-integrable functions are weakly measurable and Pettis integrable, thereby connecting the new integral to established theory.
• To demonstrate that the RL integral provides a useful tool for studying the geometry of non-separable Banach spaces.

Proposed method

• Define the upper Lebesgue integral of a non-negative function as the infimum of Lebesgue integrals of measurable majorants.
• Introduce the RL-norm via the upper Lebesgue integral of the norm of a function, forming the space $\overline{L_1}(\Omega, \Sigma, \mu, X)$.
• Define RL-integral sums using countable partitions and sampling points, and define the RL integral as the limit of such sums in the RL-norm.
• Prove that if a function has an integrable majorant in norm, then its composition with any functional is measurable, implying weak measurability.
• Use the existence of smallest measurable majorants and largest minorants of $x^*f$ to derive contradictions when assuming non-measurability, proving weak measurability.
• Show that for any measurable set $A$, the functional $x^*$ applied to the RL integral of $f|_A$ equals the Lebesgue integral of $x^*f|_A$, confirming Pettis integrability.

Experimental results

Research questions

• RQ1 Can a new vector-valued integral be defined that generalizes both Riemann and Bochner integrals while remaining more tractable than the Pettis integral?
• RQ2 Does the RL integral imply weak measurability and Pettis integrability for functions with integrable majorants?
• RQ3 How does the RL integral relate to the geometry of non-separable Banach spaces?
• RQ4 What are the structural properties of the RL space $\overline{L_1}$, and how does it compare to Bochner and Pettis integrable functions?

Key findings

• The RL integral is well-defined and forms a Banach space under the RL-norm, which generalizes both Riemann and Bochner integrals.
• Any function in $\overline{L_1}$ with an integrable majorant is weakly measurable, as shown by contradiction using measurable majorants and minorants of $x^*f$.
• The RL integral of a function $f$ over a measurable set $A$ is the unique element $x_A \in X$ such that $x^*(x_A) = \int_A x^*f \, d\mu$ for all $x^* \in X^*$, proving Pettis integrability.
• The RL integral is strictly more restrictive than the Pettis integral but avoids its pathologies, such as non-completeness and lack of differentiability of antiderivatives.
• The construction of smallest measurable majorants and largest minorants of $x^*f$ leads to a contradiction if $f$ is not weakly measurable, thus proving weak measurability.
• The RL integral provides a natural bridge between Riemann/Bochner integration and Pettis integration, making it a valuable tool in Banach space geometry, especially in non-separable settings.