Real Analysis Exchange

Change of Variable Theorems for the KH Integral

Michael Bensimhoun

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

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

References

• R. J. Bagby, The substitution theorem for Riemann integrals, Real Anal. Exchange, 27(1) (2001), 309–314.
• R. G. Bartle, A Modern Theory of Integration, (English summary) Graduate Studies in Mathematics, 32, American Mathematical Society, Providence, RI, 2001.
• M. Federson, Some peculiarities of the Henstock and Kurzweil integrals of Banach spaces-valued functions, Real Anal. Exchange, 29(1) (2003–2004), 439–460.
• H. Kestelman, Change of variable in Riemann integration, Math. Gazette, 45 (1961), 17–23.
• S. Leader, The Kurzweil-Henstock integral and its differentials, (English summary), A unified theory of integration on ${\Bbb R}$ and ${\Bbb R \sp n}$, Monographs and Textbooks in Pure and Applied Mathematics, 242, Marcel Dekker, New York, 2001.
• S. Leader, Change of variable in Kurzweil-Henstock Stieltjes integrals, Real Anal. Exchange, 29(2) (2003-2004), 905–920.
• J. Serrin and D. Varberg, A general chain rule for derivatives and the change of variables formula for the Lebesgue integral, Am. Math. Mon., 76 (1969), 514–520.
• K. R. Stromberg, Introduction to Classical Real Analysis, Wadsworth International Mathematics Series, Wadsworth International, Belmont, Calif., 1981.
• E. Talvila, The distributional Denjoy integral, Real Anal. Exchange, 33(1) (2008), 51–82.