Timeline for How necessary is the knowledge of Lebesgue integral for non-analysts? [closed]
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20 events
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| Dec 29, 2016 at 10:58 | history | closed |
Nate Eldredge Alexandre Eremenko user44143 Jan-Christoph Schlage-Puchta Stefan Kohl♦ |
Opinion-based | |
| Dec 29, 2016 at 10:12 | comment | added | Jan-Christoph Schlage-Puchta | As educational goals both the construction of the Lebesgue measure and the completion of a metric space are things one should have seen. I believe it is a bad idea to introduce the real numbers in this way to first year students (I have seen someone doing this, and students were not happy). So the real question should be: At which point in the curriculum do you teach measure and completion? The answer depends on your students, the mandatory courses, the length of your program, and other particulars of your institute. | |
| Dec 29, 2016 at 9:59 | comment | added | Jan-Christoph Schlage-Puchta | @Dirk: When I heard functional analysis, $L^p$ was introduced as an abstract completion of a metric space, then it was shown that elements of $L^p$ can be interpreted as functions almost everywhere, integral was defined by continuity, and finally the measure of a set was defined as the integral of its indicator functions. I think this is a good approach as long as you stay in the very concrete, i.e. $L^p$, $H^p$, applications to PDE and such. If you want to deal with more abstract things, the more usual approach (first the measure, then the integral) is proabably better. | |
| Dec 29, 2016 at 3:06 | review | Close votes | |||
| Dec 29, 2016 at 10:58 | |||||
| Dec 23, 2016 at 20:42 | comment | added | Alex M. | @Wojowu: If I have a topological space and consider a Borel measure on it, how am I going to integrate continuous functions without Lebesgue's theory? | |
| Dec 23, 2016 at 19:40 | comment | added | Nate Eldredge | This question seems kind of pointless to me. You're going to get some people saying "I am an algebraist and never use this", some others saying "here's this problem in algebra that uses it", someone replying "but that's not really algebra", etc, etc. And if you're trying to get a sense of what fraction of algebraists use it, that's a poll, and polls are off topic. | |
| Dec 23, 2016 at 18:17 | comment | added | Sam Hopkins | I almost never use analysis in my "professional" mathematical life. But, as I undergrad, I took a (standard, upper-level) real analysis class whose main purpose was the construction of the Lebesgue integral, and I think it was a formative experience- until that point, most proofs I had seen were of the single-paragraph kind, and then along comes this powerful, intuitive tool which nevertheless requires a lot of machinery and technical detail to develop. I see no reason to eschew this experience. | |
| Dec 23, 2016 at 17:52 | comment | added | Yemon Choi | @MattF. Speaking as an analyst with algebraic inclinations, who's seen Tom L. present that result and variants at conferences; no, it really isn't that simple. It is a very nice approach, and one to which I am quite partial at times, but it is not a panacea (nor, IIRC, has Tom ever claimed it is). The point of learning the theory of the Lebesgue integral is not to define/characterize $L^1$ as a Banach space... | |
| Dec 23, 2016 at 17:42 | comment | added | Yemon Choi | To play devil's advocate for a minute: I learned a fair bit of (soft) functional analysis without needing to use the "proper" construction of $L^2[0,1]$, and instead regarding it as "the" completion of $C[0,1]$ with respect to the natural inner product. We should also distinguish between needing to know the statement of the Dominated Convergence Theorem, and needing to know how one constructs Lebesgue measure using e.g. Caratheodory extension | |
| Dec 23, 2016 at 17:02 | comment | added | KConrad | Even people who are not analysts may need to use analysis. For example, in number theory one may need to work with infinite (Euler) products or integrate complex-valued functions on real/complex/$p$-adic/adelic topological groups with respect to Haar measure and invoke the dominated convergence theorem, so the powerful theorems about integration based on Lebesgue's ideas are definitely needed. To integrate $p$-adic valued functions on $p$-adic groups, a basic approach is through $p$-adic valued measures, which are inspired by classical measure theory. | |
| Dec 23, 2016 at 17:01 | comment | added | nfdc23 | Why would one want to encourage anyone planning to become a professional mathematician to embrace ignorance of such a basic, beautiful, and useful topic in mathematics? This isn't exactly an obscure and highly technical topic; it is standard (upper-level) undergraduate material. Anyone planning to do serious work in differential geometry should know about the regularity theorems which underlie Hodge theory, and many topics in algebra are illuminated a lot by an understanding of analogues which can be understood using analytic techniques; they should all get a well-rounded education. | |
| Dec 23, 2016 at 16:59 | history | edited | asv | CC BY-SA 3.0 |
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| Dec 23, 2016 at 16:46 | comment | added | Dirk | If you take the completion of the continuous functions w. r. t. the $L^2$ norm, it is not clear that the resulting space can still be identified with a function space. | |
| Dec 23, 2016 at 16:03 | history | edited | asv | CC BY-SA 3.0 |
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| Dec 23, 2016 at 15:30 | comment | added | user44143 | For algebraists: the Lebesgue integral is the norm in the initial element for the category of Banach spaces with unit elements and averaging operators. It's really that simple. maths.ed.ac.uk/~tl/cambridge_ct14/cambridge_ct14_talk.pdf | |
| Dec 23, 2016 at 15:15 | history | edited | asv | CC BY-SA 3.0 |
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| Dec 23, 2016 at 14:25 | comment | added | asv | @Wojowu: This is correct. But my question is whether it is possible not to teach it to professional algebraists and differential geometers. | |
| Dec 23, 2016 at 14:15 | review | Close votes | |||
| Dec 23, 2016 at 18:07 | |||||
| Dec 23, 2016 at 14:05 | comment | added | Wojowu | Lebesgue integral is not necessary as long as you only deal with continuous functions, at least that;s what I'm tempted to believe. | |
| Dec 23, 2016 at 13:51 | history | asked | asv | CC BY-SA 3.0 |