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I'm in a Real Analysis course at my school right now, and I've only just been introduced to integrals other then the Riemann integral. Some integrals from what I can tell seem to be much more generic, and have very little to do with what my previous notion of an integral was (the inverse of a derivative or the area under a curve).

So my question is what actually makes something an integral? Must it have the Riemann integral as a special case, or are the all in some loose conceptual way related to the area under a curve, or the inverse of a derivative, or am I completely missing the point?

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    $\begingroup$ So your question is in a general sense: what is the general concept or notion of an integral, some conceptual idea that, say, the Lebesgue integral, or Ito integral, etc. all share. $\endgroup$ – layman Dec 14 '17 at 23:59
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    $\begingroup$ The Lesbague integal et. al. for the most part the same concept of an the Reimann integral you used in Calculus 1. The Riemann integral starts to show some theoretical weaknesses in special cases (continuous nowhere functions, and sequences of functions for examples). These other integrals are abstractions that don't break down in these special cases. $\endgroup$ – Doug M Dec 15 '17 at 0:06
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    $\begingroup$ @DougM So they all will have the Riemann or Lesbague integral as a special case then? $\endgroup$ – Benji Altman Dec 15 '17 at 0:08
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    $\begingroup$ @DougM What is changing about the concept of an integral when you move from the Riemann integral to, say, the Riemann-Stieltjes integral? Or the Lebesgue integral? The Riemann integral is taught as a way to obtain the area under a curve (and also as a way to obtain the antiderivative of a function). The same is not true for the extension integrals. So what is the conceptual idea of integration then? What is the purpose that they all share? $\endgroup$ – layman Dec 15 '17 at 0:10
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    $\begingroup$ What you're looking for is an axiomatic approach to integration. A set of axioms that something must satisfy to be called an "integral". $\endgroup$ – user223391 Dec 15 '17 at 0:38
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This is a very good, astute question. To paraphrase, and pointedly so, the question can be "what do we want an integral to do?" Yes, for example, as in the question, what should an extended notion of integral be compatible with? Certainly all elementary things, e.g., fundamental theorem of calculus, areas under curves, and so on. But/and how can we reasonably capture all this in a rigorous way?

One way, and not the only, and not the only thing we'd want, is a sort of continuity on continuous (scalar-valued, for example) functions of compact support. An easy version of the Riemann integral shows how to integrate such functions very well, compatibly with the fundamental theorem of calculus (and, by design, with area-under-curve computations).

It is not entirely trivial to give the space of continuous, compactly-supported (scalar-valued) functions on a reasonable space (e.g., $\mathbb R$) the correct topology... but if we do so, then the Riesz-Markov-Kakutani theorem says that any continuous linear functional on that space is given by "an integral (against a positive, regular, Borel measure". Good!

And ... uniquely so...

It is certainly true, though, that this characterization does not directly address issues about differentiation in a parameter under the integral, and so on. I would claim that such issues are best addressed thinking in terms of Gelfand's and Pettis' ideas from 1930s, about "weak integrals". Not elementary, but really on-target.

If you have follow-up questions, please do. I've thought about such issues for quite a while now, as they do play a significant role in much of mathematics, even while being dubiously dismissed as "folkloric" or "it's just the definition".

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    $\begingroup$ What about integrals like the Ito integral? How do they fit into integration conceptually? I know their motivation is to get around the fact that Brownian motion is not differentiable as we try to define a notion of differential equation with randomness, but I don't see the connection conceptually between this integration and Riemann integration. $\endgroup$ – layman Dec 15 '17 at 0:27
  • $\begingroup$ Anything beyond "integration on locally compact, Hausdorff" ("physical") spaces is fraught with (provable) difficulties. Thus, Feynman stuff, as well as Ito and/or Stratonovich integrals do not easily fit into the (I think decisive) ideas about integrating functions on locally compact Hausdorff spaces. Of course we know what we wish we could have, but we should know that not all these are possible... and I myself an absolutely no expert on what compromises are useful, although knowing that they are necessary. $\endgroup$ – paul garrett Dec 15 '17 at 0:45
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Most integrals seek to "find the area under the curve". The Riemann integral does this, over an interval $[a,b]$, by chopping up the interval $[a,b]$ into consecutive pieces $a = x_0 < x_1 < \dots < x_{n-1} < x_n = b$ and approximating the function below and above on each of the pieces $[x_{i-1},x_i]$. The Riemann integral of a function is the answer you get from approximating below and above (if you don't get the same answer, we say the function is not Riemann integrable).

The Lebesgue integral is a kind of generalization of the Riemann integral in which case we approximate on arbitrary (measurable) subsets of $[a,b]$ - not necessarily consecutive subintervals like we do in the Riemann integral case. And if we can find a family subsets on which we approximate that yield closer and closer upper and lower approximations, we call the common approximation the Lebesgue integral. It turns out that a necessary and sufficient condition to be able to find subsets that yield good lower and upper approximations is measurability of the function in question.

A different type of integral, if you want to call it an integral, is a "path integral". These are actually defined by a "normal" integral (such as a Riemann integral), but path integrals do not seek to find the area under a curve. I think of them as finding a weighted, total displacement along a curve. For example, if you path integrate the function 1 along a circle, you get 0. But if you path integrate a function along a circle, but that function takes higher values only at the beginning of the path, you won't get 0 but rather something closer to the high values of the function.

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    $\begingroup$ What about Ito, Stratanovich, Skorokhod, rough etc.? $\endgroup$ – user223391 Dec 15 '17 at 0:31
  • $\begingroup$ Is the Lebesgue integral really over arbitrary measurable subsets of $[a,b]$? I was under the impression we are partitioning the range $f([a,b])$, so we are looking at measurable subsets of $[a,b]$ that looks like the preimage of a sub-interval in the range. Who is to say every measurable subset of $[a,b]$ looks like that, though? $\endgroup$ – layman Dec 15 '17 at 0:31
  • $\begingroup$ @ZacharySelk I'm not too familiar with those but it seems they measure the expected error under the curve, since the curve there is not deterministic but rather random $\endgroup$ – mathworker21 Dec 15 '17 at 0:36
  • $\begingroup$ @layman those are the measurable subsets you end up choosing to approximate by. Those are the ones that yield the good upper/lower approximations. I'm not saying every measurable subset of $[a,b]$ looks like the preimage of such a measurable subset. But the other ones don't matter, because like you said, those are the ones to approximate by $\endgroup$ – mathworker21 Dec 15 '17 at 0:37
  • $\begingroup$ As someone interested in applied mathematics, it is obvious that physicists have introduced new forms of integration and new applications of integration, in particular in the area of the calculus of variations. Incorporating electromagnetism into the classical calculus of variations framework and the more recent path integral formulation of quantum mechanics are examples. I think it would be helpful (if not popular) if more mathematicians were interested in understanding, critiquing and clarifying the nature of these newer integrals, to help find new ideas and systems of thought. $\endgroup$ – James Arathoon Dec 16 '17 at 2:32

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