A Coq Formalization of Lebesgue Integration of Nonnegative Functions

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 providing the highest confidence on the correction 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 considered as 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 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 towards the formalization of $L^p$~spaces such as Banach spaces.