From: jmccarty@sun1307.ssd.usa.alcatel.com (Mike McCarty) Subject: Re: I am losing my math ability... Date: 8 May 1999 00:38:05 GMT Newsgroups: sci.math Keywords: Different types of integrals (Riemann, Lebesgue, etc.) In article , Jeremy Boden wrote: )Why not do it via news group? )It might even raise the tone of sci.math a bit! )I would certainly like to at least "listen in" to this kind of topic. ) )-- )Jeremy Boden mailto:jeremy@jboden.demon.co.uk Well, ok, you talked me into it. I'll start with a reasonable introduction to the Lebesgue integral, and show why it is as natural and easy to understand as the Riemann, but *much* more powerful and easy to use. With the Riemann integral, we partition the X axis and sum up a bunch of rectangles. Well, with the Lebesgue integral, we do basically the same thing, but we partition the Y axis, not the X axis. Let's define a "simple" function as one which takes on only a finite number of values. You could imagine a step function. Then the various values it takes on are a sort of a partition on the Y axis. We form the inverse images of the various values it takes on, and multiply its Y value by the measure of the inverse. If you think of step functions, this is *exactly* the same procedure followed by the Riemann integral, just "turned sideways". Then we sum up all the rectangles. Now of course, a "simple" function may have really bizarre pre-images for the various values it takes on. But conceptually, they are just rectangles we're summing up. This is the Lebesgue integral for simple functions. For measurable functions which are not simple, we take lim sup L(g) over all simple g with g(x) < f(x). IOW, we "refine" the partition on Y by alowing the number of values the simple functions take on to increase without bound. What is a measurable function? Well for all measurable A in the image of f, then f^-1[A] must be a measurable set. I hope this has motivated the Lebesgue integral. If not, please feel free to ask questions. Mike -- ---- char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);} This message made from 100% recycled bits. I don't speak for Alcatel <- They make me say that. ============================================================================== From: stevel@coastside.net (Steve Leibel) Subject: Re: I am losing my math ability... Date: Fri, 07 May 1999 22:00:33 -0700 Newsgroups: sci.math In article <3733A4BD.FC0AF1CC@batnet.com>, Chas F Brown wrote: > I guess a helpful example would be an integral which can only be > evaluated as a Lebesgue intergral - mayhaps one of the funkier sets like > the Cantor "take-away-middle-third" set? > Yes, but there is a much easier example. Just take f(x) = 0 if x is irrational, f(x) = 1 if x is rational. You can see that this function is not Riemann integrable because in any partition of the x-axis, any interval contains values of x for which f(x) =0, and other values for which f(x) = 1. So the lower Riemann sum is 0 and the upper Riemann sum is 1, therefore by definition f is not Riemann integrable. However it's easy to calculate the Lebesge integral, which is 1 times the measure of the set where f is 1 (the rationals, which have measure 0), and 0 times the measure of the set where f is 0 (the irrationals, which have measure 1) = 1*0 + 0*1 = 0 + 0 = 0 f is an example of a very naturally-occuring function that is Lebesge integrable but not Riemann integrable. By the way I left math grad school many years ago and I'm enjoying this thread for the same reason you are, it's fun to remember this stuff. Steve L stevel@coastside.net ============================================================================== From: dtd@world.std.com Subject: Re: I am losing my math ability... Date: Sat, 8 May 1999 15:13:14 GMT Newsgroups: sci.math In article <37324091.B50936ED@graduate.hku.hk>, Hankel O'Fung wrote: > I wish to learn some integration theory (Lebesgue, Riemann-Stieltjes, > Lebesgue-Stieltjes, Perron), and want some books that explain the rationale > behind why some integrals are defined in the way they are so defined. In > other words, I wish to read a book which is inspiring, but not containing > only theorems and proofs. Any suggestions? i don't have any book suggestions, but i can add a little motivating rationale to mr. mccarty's intro to lebesgue integration. the central benefit of lebesgue's machinery is that lebesgue makes it easier to talk about convergence of integrable functions, so as to talk about continuity properties of the integration operator itself. clearly, if f_n --> f, we want to know that Int[f_n] --> Int[f] . lebesgue integration is better than riemann for making this jump, because the inference holds for more sequences of functions f_n . partly, this improvement comes from the fact that more functions are lebesgue-integrable than are riemann-integrable. that is, often, the reason riemann can't give you the inference is that either the f_n or the limit function f fails to be riemann-integrable. with lebesgue integration, this failure is less likely. but equally valuable are lebesgue theory's nice, easy-to- apply convergence theorems, like dominated convergence, which guarantee this inference under very general conditions. at bottom, the reason lebesgue is better for talking about the continuity of integration, is precisely that lebesgue subdivides the integrand's range, instead of its domain. this has the cost of leading us to reason "backwards" a lot, working from simple intervals in the integrand's range to arbitrarily complicated preimages of those intervals in the domain. this backwardness is part of what makes lebesgue seem difficult, just as the same backwardness makes epsilon- delta proofs of continuity seem difficult at first. but, this range-to-domain backwardness allows us to seamlessly blend integrability arguments with continuity arguments. thus, lebesgue's rather counterintuitive "backwardness" gives us a machinery for integration that interoperates noiselessly with our modern topological machinery for talking about continuity. - don davis, boston ============================================================================== From: "G. A. Edgar" Subject: Re: I am losing my math ability... Date: Fri, 07 May 1999 13:53:38 -0400 Newsgroups: sci.math In article <37324091.B50936ED@graduate.hku.hk>, Hankel O'Fung wrote: > Dear all, > > I graduated from university almost ten years ago. I don't remember much > mathematics now. Now I wish to learn some integration theory (Lebesgue, > Riemann-Stieltjes, Lebesgue-Stieltjes, Perron), and want some books that > explain the rationale behind why some integrals are defined in the way > they are so defined. In other words, I wish to read a book which is > inspiring, but not containing only theorems and proofs. Any suggestions? > > Hankel I seem to recal that Henstock's recent book on integration has a long chapter on historical and motivational considerations. But that book is written assuming quite a high level of mathematical sophistication. AUTHOR Henstock, Ralph. TITLE The General theory of integration PUBLISH INFO Oxford : Clarendon Press ; New York : Oxford University Press, 1991. DESCRIPTION xi, 262 p. ; 24 cm. SERIES Oxford mathematical monographs. R. A. Gordon's book compares and contrasts many integration methods, and is a bit more down-to-earth. But still probably not for someone who doesn't remember much mathematics. AUTHOR Gordon, Russell A., 1955- TITLE The Integrals of Lebesgue, Denjoy, Perron, and Henstock PUBLISH INFO Providence, R.I. : American Mathematical Society, c1994. DESCRIPTION xi, 395 p. : ill. ; 27 cm. SERIES Graduate studies in mathematics ; v. 4. If you omit "Perron" from your list, there are more elementary texts that discuss integration among other topics, such as Rudin, Royden, Stromberg; but motivational material may be just a paragraph at the end of a chapter. AUTHOR Rudin, Walter, 1921- TITLE Principles of mathematical analysis. PUBLISH INFO New York, McGraw-Hill, 1953. DESCRIPTION 227 p. 24 cm. SERIES International series in pure and applied mathematics. AUTHOR Royden, H. L. TITLE Real analysis. PUBLISH INFO New York, Macmillan [1963] DESCRIPTION 284 p. 24 cm. AUTHOR Stromberg, Karl Robert, 1931- TITLE Introduction to classical real analysis PUBLISH INFO Belmont, Calif. : Wadsworth International Group, c1981. DESCRIPTION ix, 575 p. ; 24 cm. -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax)