A continuous function of a Stieltjes-integrable function is Stieltjes integrable for a nondecreasing integrator

Choose $K$ with $\mathrm{\mid }\varphi \mathrm{\mid }\le K$. Given $\epsilon >0$, uniform continuity supplies $\eta >0$ such that $\mathrm{\mid }u-v\mathrm{\mid }<\eta$ implies $\mathrm{\mid }\varphi (u)-\varphi (v)\mathrm{\mid }<\epsilon \mathrm{/}(2(1+\alpha (b)-\alpha (a)))$.

By [L1], choose a partition $P$ for which ${\sum }_I{\mathrm{osc}}_I(f){\mathrm{\Delta }}_I\alpha <\eta \epsilon \mathrm{/}(4K+1)$. Split its intervals into those with ${\mathrm{osc}}_I(f)<\eta$ and the rest. The first class contributes less than $\epsilon \mathrm{/}2$ to the weighted oscillation sum of $\varphi \circ f$. In the second class, ${\mathrm{osc}}_I(\varphi \circ f)\le 2K$, while $\eta \sum {\mathrm{\Delta }}_I\alpha \le {\sum }_I{\mathrm{osc}}_I(f){\mathrm{\Delta }}_I\alpha$; hence it too contributes less than $\epsilon \mathrm{/}2$.

Thus the weighted oscillation condition in [L1] holds for $\varphi \circ f$. The same theorem says that $f$ is continuous at every discontinuity of $\alpha$; continuity of $\varphi$ makes $\varphi \circ f$ continuous there as well. Both clauses of [L1] now give $\varphi \circ f\in R(\alpha )$.