I'm new to degree level mathematics and am gradually working my way through 'Mathematical Analysis' by Apostol. I am having difficulty trying to visualise the concept of Riemann-Stieltjes integration.

Is it possible to graph / visualise the Riemann-Stieltjes sums? For example, in this post "Calculate the Riemann Stieltjes integral", the user Tao provides a link to the following document: http://math2.eku.edu/jones/analysis_e_084.pdf

The following example is calculated:

Let $f(x) = -8+6x-x^{2}$ on $[2,5]$ and let $P = \{2,2.6,3,5\}$.

If $\alpha(x) = \sqrt{x}$, then

\begin{align} RS^{+}(f,\alpha,P) &= \Sigma_{k=1}^{n} M_{k} (\alpha(x_k)-\alpha(x_{k-1})\\ &= f(2.6)(\alpha(2.6)-\alpha(2)) + f(3)(\alpha(3)-\alpha(2.6)) + f(3)(\alpha(5)-\alpha(3))\\ &= 0.7901 \end{align} and \begin{align} RS^{-}(f,\alpha,P) &= \Sigma_{k=1}^{n} M_{k} (\alpha(x_k)-\alpha(x_{k-1})\\ &= f(0)(\alpha(2.6)-\alpha(2)) + f(2.6)(\alpha(3)-\alpha(2.6)) + f(5)(\alpha(5)-\alpha(3))\\ &= -11.8039 \end{align}


I believe I understand the concepts of least upper bounds and greatest lower bounds but I'm having trouble understanding what the "width" really refers to in the above example (it seems we are taking the difference between partitions using $\sqrt{x}$, yet multiplying by lub or glb from $f(x)$?)

Is it possible to plot both $f(x)$ and $\alpha=\sqrt{x}$ and show the areas being summed? Or am I barking up the wrong tree?

Many thanks,


  • $\begingroup$ As with Riemann integrals, I think trying to visualise the upper/lower sums is not particularly edifying. Instead, one pictures a Riemann integral as an area under a curve. I think a similar approach with Riemann-Stieltjes integrals is similarly fruitful. If $\alpha(x) = \sqrt{x}$, then, loosely, you can think of $d \alpha(x) $ as ${1 \over 2 \sqrt{x}} dx$, that is, the integral $\int f(x) d \alpha(x) $ becomes $\int f(x) {1 \over 2 \sqrt{x}} dx$, so the $\alpha$ (or rather its derivative) acts as a 'weighting' function of sorts. Of course, this only helps with differentiable $\alpha$, $\endgroup$ – copper.hat Mar 2 '14 at 9:07
  • $\begingroup$ but it gives the general idea of $\alpha$ being a weight of sorts. $\endgroup$ – copper.hat Mar 2 '14 at 9:07
  • $\begingroup$ @copper.hat Much appreciated, would you be able to suggest any references which could provide additional intuition? I've seen a couple of other examples which use discontinuous functions and involve (what looks like) taking the limit from the right and left when calculating $\alpha(x)$ but I still have difficulty trying to understand what it is I'm calculating if not an 'area'? $\endgroup$ – John Smith Mar 2 '14 at 10:44
  • $\begingroup$ I found this very useful with regard to my original question: math.stackexchange.com/questions/295383/… $\endgroup$ – John Smith Mar 2 '14 at 14:03
  • $\begingroup$ All my references for Riemann-Stieltjes integration deal with the properties/analytic details and not really with intuition. You are calculating an area in some sense, but a weighted area. Maybe think of computing the expected value of some function with respect to a non-uniform probability? $\endgroup$ – copper.hat Mar 2 '14 at 16:14

For Geometric interpretation of Riemann-stieltjes integral please see "

A Geometric Interpretation of the Riemann-Stieltjes Integral

Gregory L. Bullock

The American Mathematical Monthly

Vol. 95, No. 5 (May, 1988), pp. 448-455 " It has good graphics and geometric 3 dimentional interpretation.

  • $\begingroup$ Many thanks, appreciated. $\endgroup$ – John Smith Jul 30 '15 at 7:20

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