On the existence of the Riemann-Stieltjes integral

Rafał M. Łochowski

Published 2014-03-21, updated 2015-03-05Version 2

The purpose of this note is a new proof of Young's theorem on the existence of the Riemann-Stieltjes integral when the integrand and integrator have possibly unbounded variation, but they have finite $p-$variation and $q-$variation respectively, where $p>1,$ $q>1$ and $p^{-1}+q^{-1}>1.$ Our proof will follow easily from a more general theorem formulated in terms of a functional called truncated variation.