The geometry meaning of Riemann–Stieltjes integral Maybe my question seems so strange but I want to know what is the geometry meaning of Riemann stieltjes integral ??
 A: 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.
