Riemann–Stieltjes integral

def. Riemann-Stieltjes Integral. let functions $f(x),g(x):\mathbb{R}\to \mathbb{R}$. Then the R-S integral with $f(x)$ as the integrand and $g(x)$ as the integrator is defined:

$$\begin{array}{r}{\int }_a^{b}f(x)dg(x)=\underset{n\to \infty }{\lim }\sum _{i=0}^{n}f(x_i)[g(x_{i+1})-g(x_i)]\end{array}$$

where $a=x_0\le \cdots \le x_n=b$

Visualization. The value of the integral is the shadow projection of the purple fence (=from $x-g$ plane to the intersecting line of $f(x)$ and $g(x)$). The lower the slope of $g(x)$, the denser the shadow is. See: Riemann–Stieltjes integral - Wikiwand