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Many fields in mathematics start from the "dirty" approach. In calculus we do all sort of $\epsilon$-$\delta$ stuffs, until topology gives an elegant formulation using open sets. A first course in linear algebra usually starts with defining matrices, their multiplications, determinants, etc, then the whole theory is built upon that. But if we start from vector spaces and linear operators on them, everything seems to fit in much more cleanly.

What about integrals? There are at least two approaches to define Riemann integrals: upper and lower integrals; or Riemann sums of partitions. None of them looks elegant to me. So, is there a "clean" way to develop the theory of integrals?

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    I'm not sure if that's what you're looking for, but you might be interested in measure theory.2012-04-07
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    At some point there's no way around $\varepsilon$-$\delta$-stuff! (or at least the alternatives are at least as difficult to grasp)2012-04-07
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    Actually, $\epsilon$-$\delta$ was the *clean* approach (vs. the *ad hoc* and unwarranted manipulation of infinitesimals, which would have to wait until Robinson's work to get on firm logical footing). Topology is not a "cleaning up" of $\epsilon$-$\delta$ proofs, it's a *generalization*.2012-04-07
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    What would be enlightening at this point for everybody (me included) would be that the OP explains as precisely and specifically as possible how they came to think of epsilon-delta proofs as *dirty*. This is a most serious suggestion.2012-04-07
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    If you write \epsilon-delta, with the hyphen INSIDE of $\TeX$, then the hyphen will look like a minus sign thus: $\epsilon-\delta$. But you you exclude the hyphen from $\TeX$, then it's a proper hyphen: $\epsilon$-$\delta$. One could also try \epsilon\text{-}\delta : $\epsilon\text{-}\delta$.2012-04-07
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    This question is perfectly clear to me. By "dirty" the OP means "messy" and not "not rigorous", and by "clean" he means "elegant" and not "rigorous".2012-04-13
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    @StefanWalter In case your comment is meant as an answer to mine, I note that you are merely shifting the problem to a proper definition of *messy* and *elegant*. (And please use @ to signal your comments.)2012-04-14
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    @Didier: My comment wasn't directed at you exclusively. Of course these concepts cannot be defined properly. But I think the OP has done a good job to illustrate them.2012-04-14
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    @StefanWalter I disagree. Questioning the meaning of these labels (not *concepts*) and how they are used seems mandatory to avoid the mere, probably unconscious, propagation of prejudices.2012-04-14
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    @Didier: Several answerers have already understood the question perfectly well, and I suspect that you do too. I see no point in arguing just for the sake of it. If you want to explain your point of view on what labels would be "correct" (whatever this means), please go ahead.2012-04-14
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    @StefanWalter Did they? Quote: *In any case, the answer to what you want depends heavily* (emphasized) *in what you mean by "clean"*. (But please do not put *correct* between quotation marks since I did not use the word--and for some reasons.)2012-04-14
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    @Didier: Still waiting for anything constructive from your part...2012-04-14
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    @StefanWalter While you are waiting, try to ponder this: to warn against the unconscious use of some representations carrying nontrivial implications is seen (at least in some circles) as more *constructive* than to insist that these should at no cost be questioned (or do I misrepresent your stance on this?).2012-04-14
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    @Didier: My stance: The OP chose words that most people associate with other concepts (see Arturo's comment) than he does, which led to a misunderstanding. The goal of our discussion should be to remove that misunderstanding. Once that is achieved, the question can be answered. It is not necessary to discuss whether the OP's words are incorrect or inferior (or whatever you had in mind), and I doubt that such a question is at all meaningful. However, it is not your insistence on that discussion I find unconstructive, but your failure to contribute anything to it beyond asking others to clarify,2012-04-14
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    ...,issuing vague warnings, and so on.2012-04-14
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    I side with Didier. I think that **clean** is way too subjective. Why would one say that Darboux or Riemann integration is not **clean**? What is messy about them? Miku, why don't you read Spivaks Chapert 13 in calculus. He introduces integration quite "cleanly" if you must ask.2012-05-02
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    This is a more a speculation, less an answer but... does the elegant construction of Riemann integrals from the continuation of simple step functions given above, extend to the construction of Lebesgue integrals as the continuation of countable sums of disjoint step functions? Or would that be too easy?2012-05-02
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    You may want to read [A Garden of Integrals](http://www.maa.org/ebooks/dolciani/DOL31.html).2012-05-15

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