Building intuition for Riemann-Stieltjes integral

Soft question: how can I build geometric intuition for and/or visualize the Riemann–Stieltjes integral? This is pretty easy with Riemann-integral, but in case of Riemann-Stieltjes integral I always have to rely on symbolics.

Solution 1:

The Riemann-Stieltjes integral is effectively a Riemann integral, except the integrand is "weighted" at every point according to a given function, the "integrator." In fact (and you may already know this), if the integrand is $f$ and the integrator $g$ is differentiable, then $$\int_a^b f(x)dg(x)=\int_a^b f(x)g'(x)dx.$$

If $g$ is increasing (and when I learned about the Riemann-Stieltjes integral it was required to be), then you can think of $g$ as sort of acting on the real line - and also acting on the graph of $f$. For example, if $g(x)=2x$, then $g$ acts on the real line by "stretching" by a factor of $2$. If you think of $f$ as being graphed on a rubber sheet (corresponding to a coordinate plane) which you stretch yourself by a factor of $2$, then the area under the graph of $f$ will double, corresponding to the fact that $$\int_a^b f(x)d(2x)=2\int_a^bf(x)dx.$$ If $g$ is a more complicated function, then the transformation might be harder to parse, but this is effectively what is happening.