2022/01/10 by Sylvie Boldo, François Clément, Boldo, Sylvie +3
Economics, Econometrics and Finance · Mathematics · #Advanced Banach Space Theory #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Logic in Computer Science (cs.LO) #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.2201.03242
openalex publication_date 2022/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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.