Talk:Riemann–Stieltjes integral

Redirect
I redirected Stieltjes integral here, as it seems to mean the same as "Riemann-Stieltjes integral". Here is a copy of what was in the now-redirected article. Jon Olav Vik 12:10, 4 November 2005 (UTC)

The Stieltjes integral provides a direct way of (numerically) defining an integral of the type
 * $$ \int_a^b f(x) \, d g(x) $$

without first having to convert it to
 * $$ \int_a^b f(x) \, g'(x) \, dx $$

and then integrating this converted form by means of a pre-existing, non-Stieltjes integration method.

Stieltjes integration provides a means of extending any type of integration of the form
 * $$ \int_a^b f(x) \, dx, $$

such as Riemann integration, Darboux integration, or Lebesgue integration.

Thus, the form
 * $$ \int_a^b f(x) \, d g(x) $$

can be integrated by means of Riemann-Stieltjes integration, Darboux-Stieltjes integration, or Lebesgue-Stieltjes integration. Function f is called the integrand and function g is called the integrator.

Intuition
Some more text w.r.t. intuition would be appreciated. In the introduction it states that it generalizes the Riemann integral. One line in what exact is generalized without having to go back to that definition would be nice. Suppose to see this side-by-side:
 * $$ \int_a^b f(x) \, dx, $$

And:
 * $$ \int_a^b f(x) \, d g(x) $$

Then, it would be nice to have some visualisation for different choices of g(x). And see the difference with just integrating with dx. The visualization with rows versus columns w.r.t. Lebesgue vs Riemann is really nice. It demonstrates the idea immediately. A visualisation of a function that cannot be integrated by Riemann, but can be integrated by Riemann-Stieltjes would also be appreciated. And last, but not least, a visualisation of a function that cannot be integrated by Riemann-Stieltjes, but by something more general, would be awesome. Then you really see the limitations of a method.

Andy (talk) 21:45, 4 July 2013 (UTC)

error in wikipedia page
The page states that the integral is not defined if the integrator and the integrand share a point of discontinuity. However, the proper condition is that they share a common-sided discontinuity. A counterexample to the former condition is the following: the integrator is the characteristic function of [0, 1/2] and the integrand is the characteristic function of [1/2, 1]. Integrating from 0 to 1, let P be the partition {0, 1/2, 1}. It is easy to check that the difference of the upper and lower sums over P is 0. — Preceding unsigned comment added by 74.192.26.106 (talk) 15:18, 13 December 2012 (UTC)

absolute continuity
A discussion of absolute continuity is needed on these pages. It addresses the question: when is a function the integral of its (a.e.) defined derivative?

contributions?
Considering I just wrote a paper for my real analysis class that got a perfect score from my professor, should I do anything with it? I've only begun trying to contribute to Wikipedia so I'm trying to not make any blunders. Here's a link. - Anastas5425 21:16, 23 April 2007 (UTC)

I really like the paper, it's like a more thorough version of my text book. I only have one nagging question, what can I think of the Riemann-Stieltjes integral as? You stated that the Riemann integral is just the area under the curve, which I understand. But, intuitively, since the Stieltjes integral introduces the monotonic integrator $$a(x)$$, it should not be equal to the area under the curve but actually proportional to it by $$da$$. But this is only if $$a(x)$$ is a linear function, right? - 96.60.76.205 (talk) 04:35, 19 November 2008 (UTC)

Here are some good papers on Riemann–Stieltjes integral: ; ;. Bender999 (talk) 03:20, 1 August 2009 (UTC)

P&epsilon; in the "non-generalized" integral
Should it not read:

"... such that for every &epsilon; > 0 there exists a &delta; > 0 such that for every partition P with mesh(P) < &delta;, and ... "

instead of this stuff about the yet meaningless P&epsilon;?

82.130.21.149 (talk) 13:05, 9 February 2010 (UTC)


 * I've just edited it to read as follows:
 * The "limit" is here understood in the following sense: there exists a certain number A (the value of the Riemann–Stieltjes integral) such that for every &epsilon; > 0 there exists a partition P&epsilon; such that for every partition P that refines the partition P&epsilon;, and for every choice of points ci in [xi, xi+1],
 * $$|S(P,f,g)-A| < \varepsilon. \, $$
 * "Refines" does not mean "has smaller mesh"; maybe I should make that explicit. Rather, Q refines P simply means P is a subset of Q. Michael Hardy (talk) 17:19, 9 February 2010 (UTC)
 * $$|S(P,f,g)-A| < \varepsilon. \, $$
 * "Refines" does not mean "has smaller mesh"; maybe I should make that explicit. Rather, Q refines P simply means P is a subset of Q. Michael Hardy (talk) 17:19, 9 February 2010 (UTC)
 * "Refines" does not mean "has smaller mesh"; maybe I should make that explicit. Rather, Q refines P simply means P is a subset of Q. Michael Hardy (talk) 17:19, 9 February 2010 (UTC)

OK, here's what it says now:
 * The "limit" is here understood in the following sense: there exists a certain number A (the value of the Riemann–Stieltjes integral) such that for every &epsilon; > 0 there exists a partition P&epsilon; such that for every partition P that refines the partition P&epsilon; (i.e. P&epsilon; ⊆ P), and for every choice of points ci in [xi, xi+1],
 * $$|S(P,f,g)-A| < \varepsilon. \, $$
 * $$|S(P,f,g)-A| < \varepsilon. \, $$
 * $$|S(P,f,g)-A| < \varepsilon. \, $$

Michael Hardy (talk) 17:27, 9 February 2010 (UTC)


 * I'm sorry Michael, but you are quite mistaken. I will revert the edit, and attempt to clarify by adjusting the use of the word "refines", as it seems that this has led to the initial confusion.  Sławomir Biały  (talk) 16:04, 10 February 2010 (UTC)

To reply to the OP, yes that is an equivalent way to define it. Perhaps that is clearer, I don't know. But the current approach lends itself more readily to generalization. Sławomir Biały (talk) 16:12, 10 February 2010 (UTC)

Rectifier Reference
There is no reference for the rectifier subsection, and the reader cannot find a proof. — Preceding unsigned comment added by 132.66.166.248 (talk) 16:28, 21 December 2020 (UTC)