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Absoluteness of the Riemann integral

Carlos M. Parra-Londoño, Andrés F. Uribe-Zapata

TL;DR

The article develops a generalized integration framework on Boolean algebras with finitely additive measures and proves its absoluteness across transitive models of $\mathrm{ZFC}$. It then specializes to the Riemann integral on rectangles in $\mathbb{R}^{n}$ by relating it to $\lambda^{n}|_{[a,b]}$, establishing that a bounded function $f$ defined on $[a,b]$ in an inner model $M$ has an absolute extension $g$ in any outer model $N$ with equal integral values, provided such an extension exists. The main contribution is the absoluteness of the Riemann integral between models, including a uniqueness statement up to Lebesgue measure zero, and the construction of an integral on Boolean algebras that underpins this result. The work also clarifies the limitations of absoluteness for the Lebesgue integral due to the non-absoluteness of the real numbers, emphasizing the interplay between model theory, forcing techniques, and classical analysis.

Abstract

This article explores the concept of absoluteness in the context of mathematical analysis, focusing specifically on the Riemann integral on $\mathbb{R}^{n}$. In mathematical logic, "absoluteness" refers to the invariance of the truth value of certain statements in different mathematical universes. Leveraging this idea, we investigate the conditions under which the Riemann integral on $\mathbb{R}^{n}$ remains absolute between transitive models of ZFC, the standard axiomatic system in which current mathematics is usually formalized. To this end, we develop a framework for integration on Boolean algebras with respect to finitely additive measures and show that the classical Riemann integral is a particular case of this generalized approach. Our main result establishes that the Riemann integral over rectangles in $\mathbb{R}^{n}$ is absolute in the following sense: if $M \subseteq N$ are transitive models of ZFC, $a, b \in \mathbb{R}^{n} \cap M$, and $f \colon [a, b] \to \mathbb{R}$ is a bounded function in $M$, then $f$ is Riemann integrable in $M$ if, and only if, in $N$ there exists some Riemann integrable function $g \colon [a, b] \to \mathbb{R}$ extending $f$. In this case, the values of the integrals computed in each model are the same. Furthermore, the function $g$ is unique except for a measure zero set.

Absoluteness of the Riemann integral

TL;DR

The article develops a generalized integration framework on Boolean algebras with finitely additive measures and proves its absoluteness across transitive models of . It then specializes to the Riemann integral on rectangles in by relating it to , establishing that a bounded function defined on in an inner model has an absolute extension in any outer model with equal integral values, provided such an extension exists. The main contribution is the absoluteness of the Riemann integral between models, including a uniqueness statement up to Lebesgue measure zero, and the construction of an integral on Boolean algebras that underpins this result. The work also clarifies the limitations of absoluteness for the Lebesgue integral due to the non-absoluteness of the real numbers, emphasizing the interplay between model theory, forcing techniques, and classical analysis.

Abstract

This article explores the concept of absoluteness in the context of mathematical analysis, focusing specifically on the Riemann integral on . In mathematical logic, "absoluteness" refers to the invariance of the truth value of certain statements in different mathematical universes. Leveraging this idea, we investigate the conditions under which the Riemann integral on remains absolute between transitive models of ZFC, the standard axiomatic system in which current mathematics is usually formalized. To this end, we develop a framework for integration on Boolean algebras with respect to finitely additive measures and show that the classical Riemann integral is a particular case of this generalized approach. Our main result establishes that the Riemann integral over rectangles in is absolute in the following sense: if are transitive models of ZFC, , and is a bounded function in , then is Riemann integrable in if, and only if, in there exists some Riemann integrable function extending . In this case, the values of the integrals computed in each model are the same. Furthermore, the function is unique except for a measure zero set.

Paper Structure

This paper contains 10 sections, 24 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boolean algebras
  4. Finitely additive measures on Boolean algebras
  5. Connections with filters and ultrafilters
  6. Notions of absoluteness
  7. Riemann integration on Boolean algebras
  8. Integration on Boolean algebras
  9. Riemann integration on $\mathbb{R}^{n}$
  10. Absoluteness of the Riemann integral

Key Result

Theorem 1

Let $M, N$ be transitive models of $\mathrm{ZFC}$ such that $M \subseteq N$, and $n \in \mathbb{N}$. Let $a = (a_{0}, \dots, a_{n-1})$ and $b = (b_{0} \dots, b_{n-1})$ be in $\mathbb{R}^{n} \cap M$ such that for any $i < n$, $a_{i} \leq b_{i}$, and $[a, b] \coloneqq \prod_{i < n} [a_{i}, b_{i}]$. In where the left value is the result of the Riemann integral computed in $M$, and the right one is th

Figures (1)

  • Figure 2: The graph on the right represents the situation from the perspective of $M$: if $f$ is Riemann integrable, then it is a continuous function except of a set of measure zero. On the other hand, the graph on the left illustrates the situation from the perspective of $N$: with the appearance of new real numbers in $[a, b]$, the function $f$ is no longer defined over the entire rectangle $[a, b]$.

Theorems & Definitions (54)

  • Theorem 1
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Example 2.10
  • Theorem 2.11
  • ...and 44 more