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The Mimura Integral: A Unified Framework for Riemann and Lebesgue Integration

Yoshifumi Mimura

TL;DR

The paper introduces the Mimura integral, a Riemann-style construction that simultaneously recovers outer and inner measures and yields a full measure-theoretic structure, thereby unifying measure and integration. It defines the lower and upper Mimura integrals and proves their equality characterizes Mimura integrability, ultimately showing equivalence with the Lebesgue integral by identifying the resulting measure space as the Lebesgue measure space and aligning measurable sets. The framework proceeds from a concrete partition-based extension of Riemann sums to establish essential convergence theorems (Monotone, Fatou, Dominated), step-function approximations, and extensions to real and complex-valued functions, while preserving intuitive Riemannian methods. This approach provides a pedagogically accessible bridge between Riemann and Lebesgue theories and offers a concrete alternative to the Daniell integral with full measure-theoretic content embedded in the integration construction.

Abstract

An integral on Euclidean space, equivalent to the Lebesgue integral, is constructed by extending the notion of Riemann sums. In contrast to the Henstock--Kurzweil and McShane integrals, the construction recovers the full measure-theoretic structure -- outer measure, inner measure, and measurable sets -- rather than merely reproducing integration with respect to the Lebesgue measure. Whereas the classical approach to Lebesgue theory proceeds through a two-layer framework of measure and integration, these layers are unified here into a single framework, thereby avoiding duplication. Compared with the Daniell integral, the method is more concrete and accessible, serving both as an alternative to the Riemann integral and as a natural bridge to abstract Lebesgue theory.

The Mimura Integral: A Unified Framework for Riemann and Lebesgue Integration

TL;DR

The paper introduces the Mimura integral, a Riemann-style construction that simultaneously recovers outer and inner measures and yields a full measure-theoretic structure, thereby unifying measure and integration. It defines the lower and upper Mimura integrals and proves their equality characterizes Mimura integrability, ultimately showing equivalence with the Lebesgue integral by identifying the resulting measure space as the Lebesgue measure space and aligning measurable sets. The framework proceeds from a concrete partition-based extension of Riemann sums to establish essential convergence theorems (Monotone, Fatou, Dominated), step-function approximations, and extensions to real and complex-valued functions, while preserving intuitive Riemannian methods. This approach provides a pedagogically accessible bridge between Riemann and Lebesgue theories and offers a concrete alternative to the Daniell integral with full measure-theoretic content embedded in the integration construction.

Abstract

An integral on Euclidean space, equivalent to the Lebesgue integral, is constructed by extending the notion of Riemann sums. In contrast to the Henstock--Kurzweil and McShane integrals, the construction recovers the full measure-theoretic structure -- outer measure, inner measure, and measurable sets -- rather than merely reproducing integration with respect to the Lebesgue measure. Whereas the classical approach to Lebesgue theory proceeds through a two-layer framework of measure and integration, these layers are unified here into a single framework, thereby avoiding duplication. Compared with the Daniell integral, the method is more concrete and accessible, serving both as an alternative to the Riemann integral and as a natural bridge to abstract Lebesgue theory.

Paper Structure

This paper contains 16 sections, 54 theorems, 176 equations.

Table of Contents

  1. Integrals for Nonnegative Functions
  2. Finite-Measure Sets $\mathcal{M}(\mathbb{R}^d)$
  3. Essential Riemann Criterion for $\mathcal{M}^1$-Integrability
  4. Properties of the Integral for Nonnegative Functions
  5. Measurable Sets $\mathcal{M}^*(\mathbb{R}^d)$
  6. $\sigma$-Algebra Structure and $\sigma$-Additivity
  7. Structure Theorem for Measurable Sets
  8. Step-Function Approximation
  9. Nonnegative Measurable Functions
  10. Convergence Theorems for Nonnegative Functions
  11. Integrals of Real and Complex-Valued functions
  12. Relation of Riemann and Mimura integrals
  13. Equivalence with the Lebesgue integral
  14. Identification of the Measure Space $(\mathbb{R}^d, \mathcal{M}^*(\mathbb{R}^d), \mu)$
  15. Equality of the Integrals
  16. ...and 1 more sections

Key Result

Lemma 1.5

For any open set $\mathcal{O} \subset \mathbb{R}^d$, there exists a sequence $\{I_m\}_{m=1}^{\infty} \subset \mathscr{I}(\mathbb{R}^d)$ such that In particular, $\sup\limits_{m \in \mathbb{N}}\mathrm{size}(I_m)$ can be made arbitrarily small. Conversely, for any sequence $\{I_m\}_{m=1}^{\infty} \subset \mathscr{I}(\mathbb{R}^d)$, the disjoint union is an open set.

Theorems & Definitions (139)

  • Remark
  • Definition 1.1: Open $d$-dimensional intervals, their measure, and size
  • Definition 1.2: Lebesgue null set
  • Example 1.3
  • Definition 1.4: Almost everywhere
  • Lemma 1.5: Open set decomposition
  • proof
  • Definition 1.6: Partitions of an open set by open $d$-dimensional intervals
  • Definition 1.7: Super-level and sub-level sets
  • Definition 1.8: Essential supremum and infimum
  • ...and 129 more