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The distributional Denjoy integral. (English) Zbl 1154.26011
Author’s abstract: Let \(f\) be a distribution (generalized 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 \(\int_{-\infty}^\infty f = F(\infty) - F(-\infty)\). 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.

MSC:
26A39 Denjoy and Perron integrals, other special integrals
46E15 Banach spaces of continuous, differentiable or analytic functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46G12 Measures and integration on abstract linear spaces
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