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2020 The Continuous Primitive Integral in the Plane
Erik Talvila
Real Anal. Exchange 45(2): 283-326 (2020). DOI: 10.14321/realanalexch.45.2.0283

## Abstract

An integral is defined on the plane that includes the Henstock-Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions $$F(x,y)$$ defined on the extended real plane $$[-\infty,\infty]^2$$ that vanish when $$x$$ or $$y$$ is $$-\infty$$. With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative $$\partial^2/(\partial x\partial y)$$ of this space of primitives. If $$f=\partial^2/(\partial x\partial y) F$$ then the integral over interval $$[a,b]\times [c,d] \subseteq[-\infty,\infty]^2$$ is $$\int_a^b\int_c^d f=F(a,c)+F(b,d)-F(a,d)-F(b,c)$$ and $$\int_{-\infty}^\infty \int_{-\infty}^\infty f=F(\infty,\infty)$$. The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is $${\lVert f\rVert}={\lVert F\rVert}_\infty$$ where $$F$$ is the unique primitive of $$f$$. The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both $$L^1$$ and the space of Henstock-Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in $${\lVert \cdot\rVert}_\infty$$ are also inherited by the integrable distributions. It is shown that the dual space is the functions of bounded Hardy-Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder’s inequality, second mean value theorem, Fubini’s theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in $${\mathbb R}^n$$ are sketched out.

## Citation

Erik Talvila. "The Continuous Primitive Integral in the Plane." Real Anal. Exchange 45 (2) 283 - 326, 2020. https://doi.org/10.14321/realanalexch.45.2.0283

## Information

Published: 2020
First available in Project Euclid: 30 June 2020

zbMATH: 07229049
Digital Object Identifier: 10.14321/realanalexch.45.2.0283

Subjects:
Primary: 26A39 , 46F10
Secondary: 46B42

Keywords: Alexiewicz norm , Banach Algebra , Banach lattice , Banach space , continuous primitive integral , convergence theorem , convolution , dual space , generalised function , Hardy-Krause variation , ‎Henstock--Kurzweil integral , Holder inequality , integration by parts , Schwartz distribution

Rights: Copyright © 2020 Michigan State University Press

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