A Coq Formalization of the Bochner integral

TL;DR

This paper presents a formalization of the Bochner integral in Coq, extending the proof assistant's capabilities to handle Banach space-valued functions and proving key theorems like dominated convergence.

Contribution

It introduces an original formalization of simple functions, defines Bochner integrability via dependent types, and proves fundamental theorems within Coq.

Findings

Formalization of simple functions in Coq

Proof of dominated convergence theorem

Equivalence with Lebesgue integral for nonnegative functions

Abstract

The Bochner integral is a generalization of the Lebesgue integral, for functions taking their values in a Banach space. Therefore, both its mathematical definition and its formalization in the Coq proof assistant are more challenging as we cannot rely on the properties of real numbers. Our contributions include an original formalization of simple functions, Bochner integrability defined by a dependent type, and the construction of the proof of the integrability of measurable functions under mild hypotheses (weak separability). Then, we define the Bochner integral and prove several theorems, including dominated convergence and the equivalence with an existing formalization of Lebesgue integral for nonnegative functions.