Exploring the Relationship Between Riemann and Lebesgue Integration

[rbj]

why don't you read riemann some time rbj. on the next page after defining the integral, he proves a function is integrable in his (riemann's) sense if and only if it is continuous almost everywhere, [although he does not use those words].

i don't have any Riemann on my bookshelf. I'm just an Neanderthal EE ("uzed to b i cudn't even spel enjunear, now i are one.") i got my original entry-level calculus book (a nice one by Robert Seeley called "Calculus of One & Several Variables"), another one for Diff. Eq. by Kreyszig (which conveniently omits any mention of the "Dirac delta" or "unit impulse" function), another for functional analysis (also by Kreyszig) which only uses the Kronecker delta. only my EE texts seem to need a concept of an impulse function, they identify it as $\delta(t)$, and it has properties that it is 0 everywhere but t=0, yet has an integral of 1. i know that you don't like that, but where we agree is that

$$\int_{-\infty}^{+\infty} f(t) \delta(t) dt = f(0)$$

and, i'll confess to you, ultimately in our EE use of the impulse function (i'll steer away from saying "Dirac delta", but we really mean the same thing when we EEs say "unit impulse function"), the impulse function ultimately ends up in an integral and it gets evaluated as above (maybe with a constant delay term in it). so, where it counts, i don't think we disagree. I've even seen some texts represent the "unit impulse" as the first derivative of the "unit step function" (that is EE-speak for "Heaviside function").

wonk, the issue for me really is almost semantic - what defines a function of a real variable. i don't see why it has to be different from the definition of the class of "generalized functions" that the Dirac is in. the issue, to me, is what you mean by the value of "undefined" for $\delta(0)$. if you do not allow in definition of a function that it can remember that it was created as a limit of these nascent delta functions that all had an integral of one for any width greater than zero, if a function is only allowed to know its raw mapping, given x, what then, is f(x), if that is all you can have for a "function", then i don't know how we could pass the information to it that, although it is infinitesimally thin, it still has finite area. i think that is why our engineering concept and usage of the Dirac delta does not meet with the approval of mathematicians. we engineers would like to allow the function $\delta(x)$ to be allowed to remember it how it was created from a limit of nascent delta functions, all that had an area of one, no matter how thin it is.

pedagogically, for engineering students, i am convinced that students need to be introduced to a concept called the "unit impulse function" which shares basically all of the important properties of the Dirac delta function, but am still not convinced that it does any good to distract and confuse with by pointing out what differentiates a "Schwartz distribution" or a "generalized function" which is what $\delta(t)$ is from a regular function, which are what the nascent delta functions are, just so that we don't violate the rule that "If f=g at all but a point, then the Riemann integrals of f and g are equal if they exist." it depends on how we define what a "function" is. they'll never use it and it just makes it harder to teach them about convolution, linear system theory, probability (with any decently defined distribution function), and the Nyquist/Shannon sampling theorem.

wonk, our different disciplines really do have different ways of looking at the Dirac delta function. this is in (EE) texts. it's not just me.

 

[matt grime]

But rbj, the dirac delta *is* a function, it is just *not* a function from the reals to the reals. It is a function from the Fun(R), the set of (decent - let's say lebesgue integrable) functions from R to R, to the real line. It is, moreover, a functional, which is a fancy way of saying that not ony is it a function, but it respects the fact that Fun(R) is a real vector space. It is a subtle point, one I've probably not been explicit enough in stressing. There is in one sense nothing wrong with calling it a function, as long as you don't think of it is a function from R to R.

As mathwonk will also point out on occasion, in algebraic geometry one uses morphisms which do not have to be defined everywhere. It is just that these are so special, that they deserve a different name.

 

[mathwonk]

well rbj, there are several reasons your point of view may not seem embraced here uncritically. for one, you jumped into a thread whose stated purpose was not to explore the engineers point of view on dirac delta functions, but to have an explanation of whether and how the lebesgue integral generalized the riemann integral, the history of the lebesgue integral, and how it solved problems the riemann integral does not, and how it finds usefulness today.

as far as i can see none of your posts have been addressed to any of these questions of the original poster, except negatively. just because you re not interested in learnening the distinction between lebesgue and riemann integration does not mean the original poster is not.

the second reason i posted most recently, is that you went beyond quoting statements you have read in various elementary textbooks, and began assuming that any statements you did not see there gave evidence that those unmentioned concepts were not known to riemann. so i pointed you naturally enough to the original source, where the opposite is quite clearly stated and proved.

you are not going to succeed fully in making points here by simply asserting that basic textbooks or famous persons agree with you, since some of us believe ourselves to have passed beyond the level of elementary textbooks in a few of these matters. you will actually have to argue and make your points logically. you are on safer ground when you speak from your experience as an engineer in using these ideas.

(Robert T. Seeley was my real and complex analysis instructor in grad school in 1965. He is an expert on analysis as his book on the Atiyah Singer index theorem shows, but his calculus book is just another calculus book as recall, nothing like at the level of the course he taught then.)

 

[Count Iblis]

The fact that engineers are using a formalism that can only be justified using the theory of distributions means that they are effectively using that theory. Whenever they have to make ad hoc conventions on how to deal woith the square of delta functions, they stumble on the fact that they did not properly define what they are doing.

Engineering books are actually a great way to get student interested in maths. If you do it "according to the book", they'll get bogged down in all the rigorous theorems and it will take ages before they can do anything intersting.

How does an engineer solve the differential equation:

y'' + y' + y = x^2

To find a solution, he writes it as:

(D^2 + D + 1)y = x^2 -------->

y = 1/(D^2 + D + 1) x^2 = (expand in powers of (D^2 + D))

[1 - D^2 - D + (D^2 + D)^2 + higher than second order in D terms ] x^2 =

(1-D)x^2 = x^2 - 2x

Add the homogeneous solution to find the general solution.

This is from an old engineering book which simply gives this method without proof.

 

[teleport]

i didn't think that Riemann integration could even deal with f=g almost everywhere in cases like the Dirichlet function.

Riemann function, also known as Thomae's function, a more interesting variation of the Dirichlet function, which you can guess is Riemann integrable, can also be proven to be Riemann integrable using that theorem. We know this in our class before anything related to Lebesque! (might not see Lebesque this year, not sure)

To show it is integrable you can use that fact (there might be an easier way). First you prove that for any e > 0, there are finite points x s.t. f(x) >= e. Then you can define a new function that differs from f at these finite number of points x, say g(x) = f(x) if x < e, and g(x) = 0 for x >= e. After showing g is R-integrable, by that theorem we get f is R-integrable.

Just a cool example.