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2007/2008 The Distributional Denjoy Integral
Erik Talvila
Real Anal. Exchange 33(1): 51-84 (2007/2008).


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, H\"older 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.


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Erik Talvila. "The Distributional Denjoy Integral." Real Anal. Exchange 33 (1) 51 - 84, 2007/2008.


Published: 2007/2008
First available in Project Euclid: 28 April 2008

zbMATH: 1154.26011
MathSciNet: MR2402863

Primary: 26A39 , ‎46E15 , 46F05 , ‎46G12

Keywords: Alexiewicz norm , Banach lattice , continuous primitive integral , distributional Denjoy integral , Henstock-Kurzweil integral , Schwartz distributions

Rights: Copyright © 2007 Michigan State University Press

Vol.33 • No. 1 • 2007/2008
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