Uniform convergence preserves Riemann--Stieltjes integrability and commutes with the integral.Suggest a slogan
Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing, and let \(\mathcal R(\alpha)\) denote the set of bounded functions \([a,b]\to\RR\) that are Riemann–Stieltjes integrable with respect to \(\alpha\) on \([a,b]\). Let \(f_n\in\mathcal R(\alpha)\) for \(n\geq1\), and let \(f\colon[a,b]\to\RR\). If \(f_n\to f\) uniformly on \([a,b]\), then \(f\in\mathcal R(\alpha)\), the limit \(\lim_{n\to\infty}\int_a^b f_n\,d\alpha\) exists, and \[ \lim_{n\to\infty}\int_a^b f_n\,d\alpha=\int_a^b f\,d\alpha. \] [@Rud76].