📝 SummarXiv

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.