A Coq Formalization of Lebesgue Integration of Nonnegative Functions

TL;DR

This paper presents a formal Coq proof assistant implementation of Lebesgue integration for nonnegative functions, including key theorems, enhancing confidence in mathematical and numerical applications.

Contribution

It introduces a comprehensive formalization of Lebesgue integration in Coq, including measures, simple functions, and major theorems, with design choices balancing readability and usability.

Findings

Formalization of σ-algebras, measures, and simple functions in Coq

Proof of the Beppo Levi (monotone convergence) theorem and Fatou's lemma

Foundation for formalizing L^p spaces and Banach spaces

Abstract

Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of $\sigma$-algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou's lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of $L^p$~spaces such as Banach spaces.