[Paper Review] Representations of multimeasures via multivalued Bartle-Dunford-Schwartz integral

Luisa Di Piazza, Kazimierz Musiał+1 more|arXiv (Cornell University)|Feb 19, 2021

Advanced Banach Space Theory4 citations

TL;DR

This paper introduces a selection-independent multivalued Bartle-Dunford-Schwartz (BDS) integral for scalar functions with respect to multimeasures taking values in locally convex spaces. The integral is defined via support functions, enabling Radon-Nikodým theorems for general and $d_H$-multimeasures under uniform scalar absolute continuity, domination, and subordination, with key results on integrability and density representation.

ABSTRACT

An integral for a scalar function with respect to a multimeasure $N$ taking its values in a locally convex space is introduced. The definition is independent of the selections of $N$ and is related to a functional version of the Bartle-Dunford-Schwartz integral with respect to a vector measure presented by Lewis. Its properties are studied together with its application to Radon-Nikodym theorems in order to represent as an integrable derivative the ratio of two general multimeasures or two $d_H$-multimeasures; equivalent conditions are provided in both cases.

Motivation & Objective

• To define a multivalued Bartle-Dunford-Schwartz integral independent of multimeasure selections.
• To establish Radon-Nikodým theorems for general multimeasures using uniform scalar absolute continuity, domination, and subordination.
• To extend the theory to $d_H$-multimeasures, accounting for non-additivity in support functions and Rådström embeddings.
• To characterize existence of control measures via the countable chain condition (ccc) and Banach space properties.
• To provide sufficient conditions for representing one multimeasure as an integral of another via a multivalued density function.

Proposed method

• Define the multivalued BDS integral using support functions $s(x', N)$, avoiding reliance on individual selections of the multimeasure $N$.
• Utilize Lewis’ functional version of the BDS integral as a foundation for the multivalued extension.
• Introduce uniform scalar absolute continuity, domination, and subordination as key conditions for Radon-Nikodým derivatives.
• Apply the Rådström embedding $j$ to relate multimeasure differentiation to vector measure theory, while accounting for non-additivity.
• Establish existence of control measures via the countable chain condition (ccc) and Banach space structure (e.g., $cb(X)$-multimeasures).
• Use Pettis and Gelfand integrability of multifunctions to construct examples of $d_H$-multimeasures with integral representations.

Experimental results

Research questions

• RQ1 Can a multivalued Bartle-Dunford-Schwartz integral be defined independently of multimeasure selections?
• RQ2 Under what conditions does a multimeasure $M$ admit a Radon-Nikodým derivative with respect to another multimeasure $N$?
• RQ3 How do the properties of the Rådström embedding $j \circ M$ relate to the differentiation of $M$ with respect to $N$?
• RQ4 What conditions guarantee the existence of control measures for multimeasures in locally convex spaces?
• RQ5 When can a multimeasure be represented as an integral of another via a multivalued density function?

Key findings

• The multivalued BDS integral is defined via support functions, ensuring independence from specific selections of the multimeasure.
• For general multimeasures, the existence of a Radon-Nikodým derivative is characterized by uniform scalar absolute continuity, domination, and subordination.
• For $d_H$-multimeasures, strong uniform scalar conditions (strong uniform absolute continuity, domination, and subordination) are necessary and sufficient for the existence of a Radon-Nikodým derivative of both $M$ and its Rådström embedding $j \circ M$.
• The integral representation $M(E) = \int_E \theta \, dN$ holds if and only if $f_1$ is scalarly equivalent to $\theta f_2$ and $r_1 = r_2 |\theta| \, \mu$-a.e., with $\theta$ measurable.
• The Rådström embedding $j \circ M$ is representable as a BDS integral with respect to $j \circ N$ if and only if $\theta$ is non-negative.
• Control measures exist for $cb(X)$-multimeasures if and only if the space $X$ satisfies the countable chain condition (ccc), and such measures exist for $c(X)$-valued multimeasures in certain locally convex spaces.