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2009/2010 Change of Variable Theorems for the KH Integral
Michael Bensimhoun
Real Anal. Exchange 35(1): 167-194 (2009/2010).


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.


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Michael Bensimhoun. "Change of Variable Theorems for the KH Integral." Real Anal. Exchange 35 (1) 167 - 194, 2009/2010.


Published: 2009/2010
First available in Project Euclid: 27 April 2010

zbMATH: 1204.28018
MathSciNet: MR2657294

Primary: 28B05
Secondary: 26A42 , 46G10

Keywords: ‎Banach spaces , change of variable , generalized Riemann , Henstock , integral , Integration , Kurzweil , substitution , variational equivalence

Rights: Copyright © 2009 Michigan State University Press

Vol.35 • No. 1 • 2009/2010
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