Riemann–Stieltjes integrability theorem

Riemann–Stieltjes integrability theorem: Let $f:[a,b]\to\mathbb{R}$ be continuous , and let $\alpha:[a,b]\to\mathbb{R}$ be increasing. Then the Riemann–Stieltjes integral $\int_a^b f\,d\alpha$ exists.

The Riemann–Stieltjes integral generalizes the Riemann integral (take $\alpha(x)=x$) and also encodes weighted sums (step-function $\alpha$) and distribution-function-type integrators . It is a standard bridge toward measure-theoretic integration.

Proof sketch: Continuity on $[a,b]$ implies uniform continuity of $f$. For a partition $P$, the difference between upper and lower Riemann–Stieltjes sums is bounded by a weighted oscillation : $U(f,P,\alpha)-L(f,P,\alpha)\le \sum_{i} \omega_i \bigl(\alpha(x_i)-\alpha(x_{i-1})\bigr),$ where $\omega_i=\sup_{[x_{i-1},x_i]} f-\inf_{[x_{i-1},x_i]} f$. With $\|P\|$ small, uniform continuity makes each $\omega_i$ small, and the total weight sums to $\alpha(b)-\alpha(a)$, forcing $U-L$ arbitrarily small.