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arXiv · 2509.25875

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

Yoshifumi Mimura
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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.

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Yoshifumi Mimura. 2025-09-30. The Mimura Integral: A Unified Framework for Riemann and Lebesgue Integration. https://arxiv.org/abs/2509.25875

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