Riemann's condition for Riemann--Stieltjes integrability

Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing, and let \(f\colon[a,b]\to\RR\) be bounded. For a partition \(P=\theset{a=x_0<x_1<\cdots<x_k=b}\) of \([a,b]\), put \(\Delta\alpha_i\coloneqq\alpha(x_i)-\alpha(x_{i-1})\) and \[ U(P,f,\alpha)\coloneqq\sum_{i=1}^k\qty{\sup_{[x_{i-1},x_i]}f}\Delta\alpha_i,\qquad L(P,f,\alpha)\coloneqq\sum_{i=1}^k\qty{\inf_{[x_{i-1},x_i]}f}\Delta\alpha_i . \] Then \(f\in\mathcal R(\alpha)\), that is, \(f\) is Riemann–Stieltjes integrable with respect to \(\alpha\) on \([a,b]\), if and only if for every \(\varepsilon>0\) there exists a partition \(P\) of \([a,b]\) such that \[ U(P,f,\alpha)-L(P,f,\alpha)<\varepsilon. \] [@Rud76].