Timeline for Integrals from a non-analytic point of view
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 21, 2010 at 19:15 | answer | added | G. Rodrigues | timeline score: 6 | |
| Sep 13, 2010 at 16:43 | comment | added | Dmitri Pavlov | @Theo: To define “conditionally convergent” Riemann integrals you have to perform one additional step, namely, you have to pass to a limit with respect to the length of the interval. You might as well perform this step after Lebesgue integration and obtain “conditionally convergent” Lebesgue integrals. | |
| Sep 12, 2010 at 19:22 | comment | added | Theo Johnson-Freyd | Incidentally, and picking up on @Zen Harper's comment: on R, say, there are Riemann-integrable functions that are not Lebesgue integrable. For example, Riemann integration allows for "conditionally convergent" integrals that Lebesgue integration disallows. A theory that is strictly stronger than both (but doesn't generalize well to manifolds) is called "gauge integration" (nothing to do with what the physicists call "gauge integration"). A function is Lebesgue-integrable iff it is absolutely gauge-integrable, and every (even improper) Riemann-integrable function is (properly) gauge-integrable. | |
| Sep 12, 2010 at 16:18 | answer | added | Paul Siegel | timeline score: 16 | |
| Sep 12, 2010 at 14:58 | answer | added | Dmitri Pavlov | timeline score: 19 | |
| Sep 12, 2010 at 14:16 | comment | added | Zen Harper | There is no single "natural" theory of integration (as far as I know!) They each have their strengths and weaknesses, depending on what you want them to do. For example, it would be wrong to say that the Lebesgue integral is "better" than the Riemann integral; there are definitely many natural problems where the simplicity of Riemann far outweighs the generality of Lebesgue (e.g. contour integration in Complex Analysis, integration of continuous Banach space-valued functions). So, I think you have to be more specific about what you want integrals for. | |
| Sep 12, 2010 at 7:41 | answer | added | Theo Johnson-Freyd | timeline score: 11 | |
| Sep 12, 2010 at 4:49 | comment | added | Andy Putman | I suppose you could say that an integral is a bilinear pairing between $k$-forms on a manifold and $k$-cycles that satisfies some axioms (it is a fun exercise to figure out what those axioms are), but I'm not convinced that that is a very enlightening thing to do. | |
| Sep 12, 2010 at 3:28 | comment | added | James D. Taylor | @Ricky: I forget. My last encounter with such ideas was roughly 13 years ago. But I'm looking for abstraction in a more categorical, rather than analytic, approach. | |
| Sep 12, 2010 at 3:25 | comment | added | James D. Taylor | @Will: That shows that integrals, as we already defined, work to define cohomology. But it doesn't get to the essence of what we want integrals to do (and then maybe we can define integrals to be any definition that does that?). In any case, having done number theoretic algebraic geometry for so long, I wonder what the essence of integrals (a tool I don't see very often in my work) is. What function do they provide? Is being an alternate way of defining cohomology really their only function? | |
| Sep 12, 2010 at 3:16 | comment | added | Will Jagy | en.wikipedia.org/wiki/De_Rham_cohomology | |
| Sep 12, 2010 at 3:11 | comment | added | user5810 | Do you know the Lebesgue integral just for R^n or for arbitrary measure spaces? If the former, then the latter might be what you're looking for. | |
| Sep 12, 2010 at 3:08 | history | asked | James D. Taylor | CC BY-SA 2.5 |