The geometry meaning of Riemann–Stieltjes integral

Question

Maybe my question seems so strange but I want to know what is the geometry meaning of Riemann stieltjes integral ??

Answer 1

Divide the $x$ axis into ranges in some arbitrary way by drawing a set of points on the $x$ axis. Now draw rectangles using the ranges in $x$ as the base, and nay point on the $f(x)$ being integrated corresponding so some point in each range as the height. Add up the areas of each rectangle that lies above the $x$ axis; call that $P$. Add up the areas of each rectangle that lies below the $x$ axis; call that $N$. For a tentative answer $P-N$.

If you do this a sequence of times, with smaller and smaller ranges, such that in the limit the largest range goes to zero, and:

• $P$ and $N$ are remain finite.
• The limiting value of $P-N$ does not depend on just which point you chose in each interval to get the height of the rectangle.
• The limiting value of $P-N$ would be the same no matter how you chose the ranges, as long as the largest range goes to zero.

If, for a particular function $f(x)$, those three conditions hold the $f(x)$ is Reimann-Stieltjes integrable, and the integral is that limit of $P-N$. That is, you have added up a bunch of thin vertical rectangles above the axis, and subtracted a bunch of thin vertical rectangles below the $x$ axix, to get a "total area".

So you have treated the area under the function like a fence with thin vertical slats.

Now you can imagine a function which behaves badly i the sense that in each range there will be some point which has unlimitted height. FOr example, let $f(x)$ be $x$ if $x$ is irrational, but the denominator of the reduced fraction representing $x$ if $x$ is rational. This function won't be Reimann-Stieltjes integrable, but there may be other definitions of integration under which the integral behaves quite calmly.