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2001/2002 The Substitution Theorem for Riemann Integrals
Richard J. Bagby
Real Anal. Exchange 27(1): 309-314 (2001/2002).


We give a completely elementary proof of the strongest possible version of the change-of-variables formula for the Riemann integration of an integrable function $f$: \[ \int_{a}^{b}f\circ G\left( x\right) g\left( x\right) \,dx=\int_{G\left( a\right) }^{G\left( b\right) }f\left( u\right) \,du \] whenever $g$ is Riemann integrable over $\left[ a,b\right] $, $G\left( x\right) =G\left( a\right) +\int_{a}^{x}g\left( t\right) \,dt$, and $f$ is Riemann integrable over the image of $\left[ a,b\right] $ under $G$. Our arguments do not require $f$ to be real-valued; it can take its values in an arbitrary Banach space.


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Richard J. Bagby. "The Substitution Theorem for Riemann Integrals." Real Anal. Exchange 27 (1) 309 - 314, 2001/2002.


Published: 2001/2002
First available in Project Euclid: 6 June 2008

zbMATH: 1012.26008
MathSciNet: MR1887860

Primary: 26A42

Keywords: Banach space , change-of-variable , Riemann integral , substitution

Rights: Copyright © 2001 Michigan State University Press

Vol.27 • No. 1 • 2001/2002
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