Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 1 / b / Solution

It suffices by part a to treat a nondecreasing right-continuous integrator, whose increments define a finite Lebesgue–Stieltjes measure ${\mu }_f$ on $[0,1]$. Let ${\alpha }_n$ be the left-endpoint step approximation on the dyadic intervals. The displayed sum is exactlyThe continuous function $\alpha$ is uniformly continuous on the compact interval, so $\Vert {\alpha }_n-\alpha {\Vert }_{\infty }\to 0$. ThereforeApply this separately to $g$ and $h$ to obtain the asserted Lebesgue-Stieltjes integral limit.