Timeline for Why is Lebesgue integration taught using positive and negative parts of functions?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 24, 2011 at 22:53 | vote | accept | KConrad | ||
| Jun 10, 2011 at 15:55 | comment | added | Gerald Edgar | There is an extensive theory of ordered vector spaces. Somewhat analogous to the theory of topological vector spaces. (Considerable overlap, but neither includes the other.) Nowadays not so widely known as it once was. Its heyday was 1930 to 1960, perhaps. | |
| May 20, 2010 at 22:18 | comment | added | Carl Offner | Yes, that's exactly right. If you're dealing with a metric space, everything has to be finite, so you don't want to start out dealing with simple functions that can assume infinite values. Another book that develops the integral metrically is Halmos. These two ways of looking at the integral are both important, and most treatments actually slip back and forth from one to the other. And what makes the integral such a versatile and remarkable object is its relation to different structures -- order, metric, and certainly linear as well. The fact that it is a linear operator is quite central. | |
| May 20, 2010 at 9:34 | comment | added | KConrad | Fabrizio, Carl is referring, I believe, to the L^1-functions as a metric completion in the L^1-seminorm of the space of step maps (simple maps) with real values. You could also look at the L^1-seminorm on step maps with values in a Banach space (as Lang does, with absolutely no difference in treatment, for him, from the case of real-valued functions as he makes no systematic use of monotone convergence or Fatou's lemma from the start). | |
| May 20, 2010 at 7:28 | comment | added | Fabrizio Polo | Completion of what? I understand how the extended reals can be seen as an order completion of ${\mathbb R}.$ But what does metric completion have to do with vector valued integrals? What are we metrically completing? ${\mathbb R}$ is already complete. The connection between order and monotone convergence makes sense. But the examples of the "metric" approach seem off topic. Are you saying convergence in measure and completeness of $L_p$ are more natural if you avoid the extended reals? I don't get it. | |
| May 19, 2010 at 22:51 | comment | added | KConrad | Ah, this is a nice point of view that had never explicitly occurred to me before. It's just like Dedekind cuts vs. Cauchy sequences to construct the real numbers from the rationals. Each of those has generalizations in directions not covered by the other case (e.g., ordered groups being completed using Dedekind cuts). To a newcomer Dedekind cuts require less baggage to describe, even though it might have far less potential down the road. In the case of integration, does the order-theoretic way of approaching L^1 have any application beyond the case of real-valued functions? | |
| May 19, 2010 at 1:10 | history | answered | Carl Offner | CC BY-SA 2.5 |