From: robjohn9@idt.net (Rob Johnson) Subject: Re: Riemann-Stieltjes integral Date: 29 Mar 2000 00:43:49 GMT Newsgroups: sci.math Summary: [missing] In article , aladinveliki@yahoo.com (G. Simic) wrote: >Count Riemann-Stieltjes integral (from 0 to 3) ( x)d( g(x)) >where g(x)=[x]-x, [x] is whole part from x. Breaking the domain into the three intervals, we get |\ 1 | x d(0-x) = -1/2 \| 0 |\ 2 | x d(1-x) = -3/2 \| 1 |\ 3 | x d(2-x) = -5/2 \| 2 Thus, Riemann-Stieltjes integration gives the integral to be -9/2. My guess is this is from an example of a failed case of integration by parts: |\ 3 |3 |\ 3 | g(x) dx = g(x) x | - | x dg(x) \| 0 |0 \| 0 -3/2 = [g(3) 3 - g(0) 0] - -9/2 Solving for g(3) gives g(3) = +1, but g(x) ranges from 0 to -1. The main problem is that the measure dg(x) is not absolutely continuous with respect to Lebesgue measure. It consists of point masses of weight 1 at all integers, plus an absolutely continuous part, -1 dx. It is the point masses that Riemann-Stieltjes integration cannot handle. That is why integration by parts usually requires both functions be absolutely continuous, at least until it is formulated in terms of distributions. Rob Johnson robjohn9@idt.net ============================================================================== From: "G. A. Edgar" Subject: Re: Riemann-Stieltjes integral Date: Wed, 29 Mar 2000 08:12:47 -0500 Newsgroups: sci.math In article <8brjk5$hg6@nnrp3.farm.idt.net>, Rob Johnson wrote: > Breaking the domain into the three intervals, we get > > |\ 1 > | x d(0-x) = -1/2 > \| 0 > > |\ 2 > | x d(1-x) = -3/2 > \| 1 > > |\ 3 > | x d(2-x) = -5/2 > \| 2 > > Thus, Riemann-Stieltjes integration gives the integral to be -9/2. This is incorrect. You miss the jumps. When done correctly, Riemann-Stieltjes integration CAN handle jumps. But you are also correct that in order to write integral f(x) d g(x) = integral f(x) g'(x) dx, we need g to be absolutely continuous. -- Gerald A. Edgar edgar@math.ohio-state.edu ============================================================================== From: robjohn9@idt.net (Rob Johnson) Subject: Re: Riemann-Stieltjes integral Date: 29 Mar 2000 14:37:23 GMT Newsgroups: sci.math In article <290320000812471528%edgar@math.ohio-state.edu.nospam>, "G. A. Edgar" wrote: [entire article as above --djr] Oops, my mistake. I was doing the Riemann integral and not the Stieltjes integral. Thus, I used integral f dg as a shorthand for integral f g' dx. My apologies. Rob Johnson robjohn9@idt.net