The Substitution Theorem for Riemann Integrals

Abstract

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.