Illinois Journal of Mathematics

The regulated primitive integral

Erik Talvila

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A function on the real line is called regulated if it has a left limit and a right limit at each point. If $f$ is a Schwartz distribution on the real line such that $f=F'$ (distributional or weak derivative) for a regulated function $F$ then the regulated primitive integral of $f$ is $\int_{(a,b)}f=F(b-)-F(a+)$, with similar definitions for other types of intervals. The space of integrable distributions is a Banach space and Banach lattice under the Alexiewicz norm. It contains the spaces of Lebesgue and Henstock–Kurzweil integrable functions as continuous embeddings. It is the completion of the space of signed Radon measures in the Alexiewicz norm. Functions of bounded variation form the dual space and the space of multipliers. The integrable distributions are a module over the functions of bounded variation. Properties such as integration by parts, change of variables, Hölder inequality, Taylor’s theorem and convergence theorems are proved.

Article information

Illinois J. Math., Volume 53, Number 4 (2009), 1187-1219.

First available in Project Euclid: 22 November 2010

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Zentralblatt MATH identifier

Primary: 26A39: Denjoy and Perron integrals, other special integrals 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]
Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions 46F05: Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35]


Talvila, Erik. The regulated primitive integral. Illinois J. Math. 53 (2009), no. 4, 1187--1219. doi:10.1215/ijm/1290435346.

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