The Radon-Nikodým Theorem and the Lebesgue-Stieltjes Measure in Coq

Abstract

We are concerned with the formalization of measure theory in the Coq proof assistant. Concretely, we extend MathComp-Analysis, a library for functional analysis built on top of the Mathematical Components library, with standard constructions such as charges and the Lebesgue-Stieltjes measure, and with standard theorems such as the Hahn decomposition theorem and the Radon-Nikodým theorem. These are prerequisites for the formalization of probabilistic programs, of probability theory, and also for other applications such as the formalization of connections between derivatives and integrals.