Riemann–Stieltjes integrability theorem

Riemann–Stieltjes integrability theorem: Let $a<b$. If $f:[a,b]\to\mathbb{R}$ is continuous and $g:[a,b]\to\mathbb{R}$ is a bounded variation function , then $f$ is Riemann–Stieltjes integrable with respect to the integrator function $g$ on $[a,b]$.

Since every monotone function has bounded variation, this theorem extends existence of the Riemann integral (the case $g(x)=x$) to a wider class of integrators.