Real Analysis Exchange

Change of Variable Theorems for the KH Integral

Michael Bensimhoun

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Abstract

Let $f : [a,b] \subseteq \bar{\mathbb{R}}\to \mathcal{E}$ and $\phi : [a,b]\to \mathcal{F}$, where $(\mathcal{E},\mathcal{F},\mathcal{G})$ is a Banach space triple. a) We prove that if $\phi$ is continuous $[c,d ]\to [a,b]$ and $f\circ \psi \cdot d\phi\circ \psi$ is Kurzweil or Henstock variationally integrable, then so is $f\cdot d\phi$ and fulfills the well known change of variable formula. It follows that if $\psi$ is an indefinite Henstock integral and if $f\circ\psi\, \psi' dx $ is K-H integrable, then so is $f dx$ and the change of variable formula applies. b) We produce several versions of the converse of a), that is, we give necessary and sufficient conditions in order that with $\psi$ as above, the integrability of $f\cdot d\phi$ implies that of $f\circ \psi \cdot d\phi\circ\psi$ and the change of variable formula.

Article information

Source
Real Anal. Exchange, Volume 35, Number 1 (2009), 167-194.

Dates
First available in Project Euclid: 27 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.rae/1272376230

Mathematical Reviews number (MathSciNet)
MR2657294

Zentralblatt MATH identifier
1204.28018

Subjects
Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10]
Secondary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]

Keywords
integral integration Kurzweil Henstock generalized Riemann substitution change of variable variational equivalence Banach spaces

Citation

Bensimhoun, Michael. Change of Variable Theorems for the KH Integral. Real Anal. Exchange 35 (2009), no. 1, 167--194. https://projecteuclid.org/euclid.rae/1272376230


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