Timeline for Why is Lebesgue integration taught using positive and negative parts of functions?
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8 events
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| Aug 10, 2010 at 19:24 | comment | added | G. Rodrigues | (continued) And these integrals coming from finitely additive measures are not totally useless or that exotic. Example: spectral measures are not (strongly) $\sigma$-additive. As for the usefulness, let me just remark that via them, a snappy proof of (a slight variant of the) Riesz representation theorem can be gotten. | |
| Aug 10, 2010 at 19:22 | comment | added | G. Rodrigues | I will not comment about the supposed pedagogical advantages of each method, but I for one agree completely with KConrad that the metric completion approach is far more preferable (and definitely is not longer). Whatever route you take, the infinite measure case needs special handling, anyway. I would also like to note that the equality $\int h df^{\ast}\mu= \int hf d\mu$ is a formal consequence of the universal property of the spaces of integrable functions. It's even true for the integral produced by any finitely additive bounded measure on a Boolean algebra with values in a Banach space! | |
| May 20, 2010 at 9:48 | comment | added | KConrad | Trying to get somewhere quickly is not a point I had thought about. I assumed that students who study Lebesgue integration will already know about completions of metric spaces, so the vector-valued approach (even if used exclusively for real-valued functions in the course) should be accessible, although admittedly it may take more time than the traditional way to reach key theorems. Trying to teach with the vector-valued approach while also doing things specific to the real-valued case (ex: monotone conv. thm) which require infty as a function value could create confusion. Спасибо за ответ. | |
| May 20, 2010 at 9:20 | comment | added | Sergei Ivanov | When I was a student, the apparent ugliness of this approach striked me too. I tried to find a better way but failed: everything had serious disadvantages from undergraduate teaching perspective. | |
| May 20, 2010 at 8:49 | comment | added | Sergei Ivanov | My point is that this approach has pedagogical advantages: a student gets familiar with a useful technique while working on this definition. Another plus is that this definition is short. | |
| May 19, 2010 at 23:05 | comment | added | KConrad | When I said in my previous comment that the vector-valued treatment builds on the non-negative case, I meant that in a negative sense. It seems weird to create L^1 for real-valued functions in a fundamentally different way than you do for vector-valued (i.e., Banach space valued) functions. | |
| May 19, 2010 at 22:57 | comment | added | KConrad | I agree the non-negative case deserves a separate treatment because of its features which don't work in the general case (function values not being in the real numbers), just like positive measures have their own features which don't occur in other settings like complex measures. But usually the vector-valued treatment builds on this non-negative case rather than developing it from scratch to see it runs on its own internal logic. | |
| May 19, 2010 at 8:21 | history | answered | Sergei Ivanov | CC BY-SA 2.5 |