Timeline for Why is Lebesgue integration taught using positive and negative parts of functions?
Current License: CC BY-SA 2.5
8 events
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| Jun 10, 2011 at 15:59 | comment | added | Gerald Edgar | For the Riemann integral, there are still two basic approaches. One using "upper" and "lower" sums, based on order. The other using Riemann sums (or Darboux sums) with convergence obtained from some sort of Cauchy criterion. That (complicated, non-sequential) type of convergence is what Moore and Smith were doing when they invented nets. | |
| May 19, 2010 at 22:46 | comment | added | KConrad | Andrea, when first developing the Riemann integral you don't do them on the whole real line. That only comes later, with the definition being in terms of the integration on bounded intervals (or, say, continuous functions) which have already been defined by more basic methods. This is what I had in mind when I said improper integrals are not part of the basic development. By "basic" I meant "initial". It comes later. | |
| May 19, 2010 at 18:48 | comment | added | Victor Protsak | Indeed, most functions in the first course aren't integrable over R anyway! My point was, there is no need to integrate smooth functions over R. The natural condition for being able to do it is that the functions be compactly supported. The one exception I can think of is the gaussian density, but if you are doing probability theory, you need Lebesgue integration anyway. | |
| May 19, 2010 at 8:30 | comment | added | Benoît Kloeckner | @Victor Protsak: for a first course, if you want to integrate some smooth examples on $\mathbb{R}$, it is much easier not to ask they are compactly supported. | |
| May 19, 2010 at 4:48 | comment | added | Victor Protsak | If I may intercede: sure you can, the function just needs to be compactly supported! This illustrates a functional (unintended pun) distinction between the Riemann integral and the Lebesgue integral: the former is a differential geometric tool and the latter is an analytic tool. | |
| May 18, 2010 at 23:11 | comment | added | Andrea Ferretti | Uhm... but this way you cannot even integrate a function on $\mathbb{R}$! | |
| May 18, 2010 at 20:57 | comment | added | KConrad | I consider the improper/principal value integrals not to be part of the basic development but something introduced only later. | |
| May 18, 2010 at 20:26 | history | answered | Andrea Ferretti | CC BY-SA 2.5 |