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

Choose $K$ with $|\phi|\le K$. Given $\varepsilon>0$, uniform continuity supplies $\eta>0$ such that $|u-v|<\eta$ implies $|\phi(u)-\phi(v)|<\varepsilon/(2(1+\alpha(b)-\alpha(a)))$.

By [L1], choose a partition $P$ for which $\sum_I\operatorname{osc}_I(f)\,\Delta_I\alpha<\eta\varepsilon/(4K+1)$. Split its intervals into those with $\operatorname{osc}_I(f)<\eta$ and the rest. The first class contributes less than $\varepsilon/2$ to the weighted oscillation sum of $\phi\circ f$. In the second class, $\operatorname{osc}_I(\phi\circ f)\le2K$, while $\eta\sum\Delta_I\alpha\le\sum_I\operatorname{osc}_I(f)\Delta_I\alpha$; hence it too contributes less than $\varepsilon/2$.

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