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Some Properties of The Finitely Additive Vector Integral

Gianluca Cassese

TL;DR

This paper develops a broad framework for representing finitely additive vector integrals (Bochner and Pettis) via representations (\tilde{m},\tilde{f}) relative to families of transformations, leveraging uniform structures, Stone/Čech techniques, and Radon–Nikodým-type properties. It establishes general existence results, specialized representations on Countable sets like \mathbb{N}, and conditions under which countably additive representations arise, particularly when the target space has the Radon–Nikodým property or when Pettis integrability holds. It also introduces an abstract standard representation with a triple (\chi,\lambda,\tilde{f}) and a standard vector measure, and explores the Pettis integrability property, including a decomposition into representable and non-representable parts. Finally, the work extends Choquet representation results to noncompact, nonconvex settings, showing how vector-valued integration interacts with dentability and PIP. Overall, the paper clarifies when and how finitely additive vector integrals admit rich, countably additive representations and Choquet-type decompositions, with implications for convergence of vector-valued martingales and Choquet-type theorems.

Abstract

We prove some results concerning the finitely additive, vector integral of Bochner and Pettis and their representation over a countably additive probability space. Applications to convergence of vector valued martingales and to the non compact Choquet theorem are provided.

Some Properties of The Finitely Additive Vector Integral

TL;DR

This paper develops a broad framework for representing finitely additive vector integrals (Bochner and Pettis) via representations (\tilde{m},\tilde{f}) relative to families of transformations, leveraging uniform structures, Stone/Čech techniques, and Radon–Nikodým-type properties. It establishes general existence results, specialized representations on Countable sets like \mathbb{N}, and conditions under which countably additive representations arise, particularly when the target space has the Radon–Nikodým property or when Pettis integrability holds. It also introduces an abstract standard representation with a triple (\chi,\lambda,\tilde{f}) and a standard vector measure, and explores the Pettis integrability property, including a decomposition into representable and non-representable parts. Finally, the work extends Choquet representation results to noncompact, nonconvex settings, showing how vector-valued integration interacts with dentability and PIP. Overall, the paper clarifies when and how finitely additive vector integrals admit rich, countably additive representations and Choquet-type decompositions, with implications for convergence of vector-valued martingales and Choquet-type theorems.

Abstract

We prove some results concerning the finitely additive, vector integral of Bochner and Pettis and their representation over a countably additive probability space. Applications to convergence of vector valued martingales and to the non compact Choquet theorem are provided.

Paper Structure

This paper contains 8 sections, 15 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A general theorem on representations
  4. Representations over $\mathbb{N}$.
  5. Abstract standard representations.
  6. The Radon-Nikodým property.
  7. The Pettis integrability property.
  8. A non compact and non convex Choquet theorem

Key Result

Theorem 1

Let $\mathscr D$ be a diagonal uniformity on $X$ and $\mathcal{H}=\mathscr{C}_{_{u}}((X,\mathscr D))\cap\mathscr L(m,f)$. Let $S$ be a non empty set, $\mathcal{N}$ an ideal of its subsets and $\tilde{f}\in\mathfrak{F}_{_{}}(S,X)$. Consider the following properties: Then (pro cover)$\Rightarrow$(pro represent) and, if $\mathscr D$ is the weak uniformity generated by $\mathcal{H}$, (pro represent)$

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Corollary 2
  • proof
  • ...and 26 more