Real Analysis Exchange

Change of Variable Theorems for the KH Integral

Michael Bensimhoun

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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

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

First available in Project Euclid: 27 April 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


Bensimhoun, Michael. Change of Variable Theorems for the KH Integral. Real Anal. Exchange 35 (2009), no. 1, 167--194.

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  • 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.