Today I want to give a modification of the Riemann integral which helps give insight into the change of variables formula.
So, we defined the Riemann integral
to be the limit as we refined the tagged partition 
of the Riemann sum
But why did we multiply by 
? Well, that was the width of a rectangular strip we were using to approximate part of the area under the graph of 
. But why should we automatically use that difference as the “width”?
Let’s imagine we’re walking past a fence. Sometimes we walk faster, and sometimes we walk slower, but at time 
we can measure the height of the fence right next to us: 
. So what’s the area of the fence? If we just integrated we’d get the wrong answer. The samples we made when walking fast made fat rectangles, while the samples we made when we were walking slowly got paired with skinny rectangles, but we gave them the same weight if they took the same time to get through that segment of the partition. We need to reweight our sums to compensate for how fast we’re walking!
Okay, so how wide should we make the rectangles? Let’s say that at time we’re at position 
along the fence. Then in the segment of the partition between times 
and 
we move from position 
to position 
, so we should make the width come out to 
. We’ll put this into our formalism from before and get the “Riemann-Stieltjes sum”:
And now we can take the limit over tagged partitions as before to get the “Riemann-Stieltjes integral”:
![\displaystyle\int\limits_{\left[a,b\right]}f(x)d\alpha(x)=\int\limits_a^bf(x)d\alpha(x)](http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle%5Cint%5Climits_%7B%5Cleft%5Ba%2Cb%5Cright%5D%7Df%28x%29d%5Calpha%28x%29%3D%5Cint%5Climits_a%5Ebf%28x%29d%5Calpha%28x%29&bg=e6e6e6&fg=000000&s=0 "\displaystyle\int\limits_{\left[a,b\right]}f(x)d\alpha(x)=\int\limits_a^bf(x)d\alpha(x)")
if this limit exists.
Here we call the function the “integrand”, and the function 
the “integrator”. Clearly, the old Riemann integral is the special case when 
.
Immediately from the definition we can see the same “additivity” (using signed intervals) in the region of integration that the Riemann integral had:
![\displaystyle\int\limits_{\left[x_1,x_3\right]+\left[x_3,x_2\right]}f(x)d\alpha(x)=\int\limits_{\left[x_1,x_3\right]}f(x)d\alpha(x)+\int\limits_{\left[x_3,x_2\right]}f(x)d\alpha(x)](http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_3%5Cright%5D%2B%5Cleft%5Bx_3%2Cx_2%5Cright%5D%7Df%28x%29d%5Calpha%28x%29%3D%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_3%5Cright%5D%7Df%28x%29d%5Calpha%28x%29%2B%5Cint%5Climits_%7B%5Cleft%5Bx_3%2Cx_2%5Cright%5D%7Df%28x%29d%5Calpha%28x%29&bg=e6e6e6&fg=000000&s=0 "\displaystyle\int\limits_{\left[x_1,x_3\right]+\left[x_3,x_2\right]}f(x)d\alpha(x)=\int\limits_{\left[x_1,x_3\right]}f(x)d\alpha(x)+\int\limits_{\left[x_3,x_2\right]}f(x)d\alpha(x)")
and the same linearity in the integrand:
![\displaystyle\int\limits_{\left[x_1,x_2\right]}af(x)+bg(x)d\alpha(x)=a\int\limits_{\left[x_1,x_2\right]}f(x)d\alpha(x)+b\int\limits_{\left[x_1,x_2\right]}g(x)d\alpha(x)](http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_2%5Cright%5D%7Daf%28x%29%2Bbg%28x%29d%5Calpha%28x%29%3Da%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_2%5Cright%5D%7Df%28x%29d%5Calpha%28x%29%2Bb%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_2%5Cright%5D%7Dg%28x%29d%5Calpha%28x%29&bg=e6e6e6&fg=000000&s=0 "\displaystyle\int\limits_{\left[x_1,x_2\right]}af(x)+bg(x)d\alpha(x)=a\int\limits_{\left[x_1,x_2\right]}f(x)d\alpha(x)+b\int\limits_{\left[x_1,x_2\right]}g(x)d\alpha(x)")
and also a new linearity in the integrator:
![\displaystyle\int\limits_{\left[x_1,x_2\right]}f(x)d(a\alpha(x)+b\beta(x))=a\int\limits_{\left[x_1,x_2\right]}f(x)d\alpha(x)+b\int\limits_{\left[x_1,x_2\right]}f(x)d\beta(x)](http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_2%5Cright%5D%7Df%28x%29d%28a%5Calpha%28x%29%2Bb%5Cbeta%28x%29%29%3Da%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_2%5Cright%5D%7Df%28x%29d%5Calpha%28x%29%2Bb%5Cint%5Climits_%7B%5Cleft%5Bx_1%2Cx_2%5Cright%5D%7Df%28x%29d%5Cbeta%28x%29&bg=e6e6e6&fg=000000&s=0 "\displaystyle\int\limits_{\left[x_1,x_2\right]}f(x)d(a\alpha(x)+b\beta(x))=a\int\limits_{\left[x_1,x_2\right]}f(x)d\alpha(x)+b\int\limits_{\left[x_1,x_2\right]}f(x)d\beta(x)")
Neat!
February 28, 2008 -
Posted by
John Armstrong |
Analysis, Calculus |
|
15 Comments
[...] follow on yesterday’s discussion of the Riemann-Stieltjes integral by looking at a restricted sort of integrator. We’ll assume here that is continuously [...]
[...] Riemann-Stieltjes Integral IV Let’s do one more easy application of the Riemann-Stieltjes integral. We know from last Friday that when our integrator is continuously differentiable, we can reduce to [...]
[...] of Bounded Variation I In our coverage of the Riemann-Stieltjes integral, we have to talk about Riemann-Stieltjes sums, which are of the [...]
Hi,
I just wanted to say thanks for the Riemann-Stieltjes pieces. I am taking second semester analysis which heavily emphasizes this integral - along with doing every proof in Rudin in class (with a take no prisoner’s attitude to the homework - i.e. NO partial credit). I also would like to say that the Professor is Dan Oberlin (FSU math) who not only is the greatest math teacher I have ever had but is a man who has so much respect for mathematics that he refuses to let it be watered down into hand-waving. His is a great teacher and even injects some humor into every class (I know this sound weird in an analysis class) - for example - we wanted to sup over a bunch of terms so we came up with “Zupping” - umlaut over the ‘u’ - which we might submit to Colbert. For all of you out there that think you are ’scooting’ - get hold of a teacher like Dan - it will blow your socks off in terms of how much math you can learn and do.
[...] If we want our Riemann-Stieltjes sums to converge to some value, we’d better have our upper and lower sums converge to that value [...]
[...] of Bounded Variation Today I want to start considering Riemann-Stieltjes integrals where the integrator is a function of bounded [...]
[...] about to do, I’m going to need a couple results about increasing integrators, and how Riemann-Stieltjes integrals with respect to them play nicely with order properties of the real [...]
[...] of integrable functions From the linearity of the Riemann-Stieltjes integral in the integrand, we know that the collection of functions that are integrable with respect to a [...]
[...] Let’s consider some conditions under which we’ll know that a given Riemann-Stieltjes integral will exist. First off, we have a straightforward adaptation of our old result that continuous [...]
[...] Function Integrators Now that we know how a Riemann-Stieltjes integral behaves where the integrand has a jump, we can put jumps together into more complicated functions. [...]
[...] Integrals I We’ve dealt with Riemann integrals and their extensions to Riemann-Stieltjes integrals. But these are both defined to integrate a function over a finite interval. What if we want to [...]
Dear Professor Armstrong,
I do have found your series of posts covering the Riemann-Stieltjes integral VERY useful to me. By using the “PDF Creator” I have already converted them into pdf files for my private use only and further study, because I see them as a reliable source and in a level appropriate to me.
I dare ask you if you intend to post a list of symbols and notation you use here. Although your notation is the standard one (for american mathematicians) as far as I have seen, it might help us in the topics we are not familiar with. To me, for instance, Abstract Algebra.
Américo Tavares
(retired engineer interested in Mathematics)
I’m not sure what I would put on such a list. I try to explain any symbols or notation as I introduce them. Are there any in particular you’re having difficulty with?
Thanks for your reply!
I think of it as a list that would start with relative few symbols and that would be extended to show new ones, when you find you have posted anything not yet there.
In principle that would require a separate page or post regularly updated.
Behind the symbols what really is important are the definitions. But that would help, anyway, I think.
For the purpose of giving you a particular example I went to one of your posts (http://unapologetic.wordpress.com/2007/02/15/cosets-and-quotients/) categorized as “Subgroups and Quotients Groups” by clicking this sub-category. There I found, for instance, among several algebraic symbols (e.g. the subgroup {e,(12)}) the “coset” concept that I still do not know what is it. This means only that I should studied the subject.
I searched for this word and found the posts where it appears.
I do realize that one cannot learn the new symbols without the associated definitions.
After being constructed this list would be very similar with a formal list of symbols in a textbook, the main difference would be that it would cover the symbols of your blog instead of the textbook.
[...] Subtracting off the integral of we get our result. (Technically to do this, we need to extend the linearity properties of Riemann-Stieltjes integrals to improper integrals, but this is [...]