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.