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A survey on the Riemann-Lebesgue integrability in non-additive setting

Anca Croitoru, Alina Gavrilut, Alina Iosif, Anna Rita Sambucini

TL;DR

This survey synthesizes the theory of Riemann-Lebesgue integration with respect to non-additive set functions, unifying scalar and vector cases, and extending to interval-valued multifunctions. It systematically introduces definitions, basic properties, and comparisons with Gould and Birkhoff integrals under finite variation, and develops convergence theorems including Fatou-type and Lebesgue-type results. A central theme is how RL integrability behaves without additivity, and how end-point representations yield interval-valued RL integrals with corresponding convergence results, including atomwise behavior. The work highlights the relevance of these generalized integrals for applications in decision making, economics, and data-driven modeling where additivity and exact probabilities are not guaranteed.

Abstract

We present in this survey some results regarding Riemann_Lebesgue integrability with respect to arbitrary non-additive set functions.

A survey on the Riemann-Lebesgue integrability in non-additive setting

TL;DR

This survey synthesizes the theory of Riemann-Lebesgue integration with respect to non-additive set functions, unifying scalar and vector cases, and extending to interval-valued multifunctions. It systematically introduces definitions, basic properties, and comparisons with Gould and Birkhoff integrals under finite variation, and develops convergence theorems including Fatou-type and Lebesgue-type results. A central theme is how RL integrability behaves without additivity, and how end-point representations yield interval-valued RL integrals with corresponding convergence results, including atomwise behavior. The work highlights the relevance of these generalized integrals for applications in decision making, economics, and data-driven modeling where additivity and exact probabilities are not guaranteed.

Abstract

We present in this survey some results regarding Riemann_Lebesgue integrability with respect to arbitrary non-additive set functions.
Paper Structure (11 sections, 23 theorems, 70 equations)

This paper contains 11 sections, 23 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic facts
  3. The Riemann-Lebesgue integrability
  4. Some properties of $RL$ $\nu$-integrability
  5. Comparison with other types of integrability
  6. Convergence results
  7. Hölder and Minkowski type inequalities
  8. Interval-valued Riemann-Lebesgue integral
  9. The interval-valued RL-integral and its properties
  10. Convergence results
  11. Convergence results on atoms

Key Result

Theorem 3.3

Let $g,h \in |RL|^1_{\nu} (X)$ and $\alpha, \beta \in\mathbb{R}$. Then: ( The same holds for $RL$-integrability ). Moreover, by ccgis, Similar results also hold for the $RL$$\nu$-integrability.

Theorems & Definitions (50)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 40 more