Real Analysis Exchange

The Distributional Denjoy Integral

Erik Talvila

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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.

Article information

Real Anal. Exchange Volume 33, Number 1 (2007), 51-84.

First available: 28 April 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 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 [See also 46E10, 46E35] 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]

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


Talvila, Erik. The Distributional Denjoy Integral. Real Analysis Exchange 33 (2007), no. 1, 51--84.

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