2006/06/21 by Erik Talvila, Talvila, Erik · 1 citation
Economics, Econometrics and Finance · Mathematics · #26A39 #46E15 #46F05 #46G12 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Stochastic processes and financial applications #math.CA #msc:26A39 #msc:46E15 #msc:46F05 #msc:46G12
paper · pdf · doi:10.48550/arxiv.math/0606537
To appear in Real Analysis Exchange
openalex publication_date 2006/06/21 · arxiv created 2007/12/07 · arxiv updated 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f be a distribution (generalised function) on the real line. If there is a continuous function F with real limits at infinity such that F'=f (distributional derivative) then the distributional integral of f is defined as ∫-∞^∞ f = F(∞) - F(-∞). It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock--Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, Hölder inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy's descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed.