A "clean" approach to integrals.

Question

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?

Answer 1

Let me say first that I fully agree with the comments posted above.

In any case, the answer to what you want depends heavily in what you mean by "clean" and, more importantly, what you mean by "theory of integrals".

Regarding "clean", your example is open sets as opposed to $\varepsilon$-$\delta$. But when you want to do concrete manipulations with open sets in a metric space you end up dealing with balls, and you prove things exactly by using $\varepsilon$-$\delta$; so it's not "cleaner" at all. Just more general, as Arturo said.

Regarding the "theory of integrals": if you want to obtain an integral that represents what the Riemann integral represents (i.e. "the area below the curve") you will have to deal with some kind of limits of sums. I personally prefer Lebesgue's approach, which is cleaner and more general, but it still implies a fair amount of "dirty".

If you dispense with the need of your integral being "the area below the curve", you might notice that all an integral does is to assign a number to a function in a linear way. So you might think of a locally compact Hausdorff space $X$, consider $C_c(X)$, i.e. the complex-valued compactly supported continuous functions on $X$, and think of all the functionals (i.e. scalar valued, linear functions) $$ \varphi:C_c(X)\to\mathbb{C}. $$ This space is well-understood, and every such functional is an "integral" in some proper sense (the Riemann integral being just a very particular example in the case where $X$ is a closed hypercube in $\mathbb{R}^n$). This last point of view is a lot "cleaner" to my taste, but it is probably not what you are looking for.

Answer 2

One can understand integration as a way of forming means. A very elegant characterization of the integral on $[0,1]$ can be obtained by using some category theory and functional analysis. You can find the characterization in A universal Banach space by Tom Leinster. I don't think the construction gives you much insight though and showing that the object actually exists will still take some time. Also, you don't actually integrate functions but certain equivalence classes of functions in this approach.

[2015-09-07: updated link]

Answer 3

AFAIK, a "polished up" version of the theory of Riemann integration can be crafted by means of the topological concept of net, or generalized sequence (see Moore-Smith Convergence on Wikipedia).

It goes on the following lines: take a function $f\colon [a, b]\to \mathbb{R}$ and introduce the set $S$ consisting of all pairs $(\mathcal{D}, \xi)$, where:

1. $\mathcal{D}=\{[x_1, x_2]\ldots[x_{\omega}, x_{\omega+1}]\}$ is a partition of $[a, b]$;
2. $\xi=\{\xi_1 \in [x_1, x_2], \ldots \xi_{\omega}\in [x_{\omega}, x_{\omega+1}]\}$ is a choice of points in $[a, b]$ subordinate to the partition $\mathcal{D}$.

For every pair $(\mathcal{D}, \xi)\in S$ we define the corresponding integral sum:

$$\sigma_f(\mathcal{D}, \xi)=\sum_{r=1}^\omega f(\xi_r)(x_{r+1}-x_r).$$

$f$ is then said to be integrable if $\sigma_f(\mathcal{D}, \xi)$ is convergent as the partition norm $\lVert \mathcal{D}\rVert=\max(x_{r+1}-x_r)$ approaches zero, a phrase which is given precise meaning in Moore-Smith's theory of convergence. If this is the case, then we can define

$$\int_a^b f(x)\, dx=\lim_{\lVert \mathcal{D}\rVert \to 0} \sigma_f(\mathcal{D}, \xi).$$

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I believe that this construction is exactly what the OP was looking for. Unfortunately the only reference I know is Italian: Analisi matematica 1 by Antonio Avantaggiati, Ambrosiana editrice, 1994.

EDIT The construction can also be found in Kelley's book General Topology: it is Problem H of chapter 2. (Thanks to Michael Greinecker for pointing this out).

Answer 4

I don't like very much the distinction clean/dirty which most of time hides a preference for formalism rather than constructivism. And I'm not sure that formalism is the best thing that happened to mathematics. For you, is cleanness just encapsulation or different approach ?

Anyway, the following construction of the Riemann integral should satisfy you.

1. Let $I$ be a segment of $\mathbb R$. Consider the vector space $\mathcal C(I)$ of functions $I\to \mathbb R$, together with the norm $\| \cdot \|_\infty$. It is a Banach space (i.e. it is complete for this norm).

2. Now, consider $V$ the sub-vector-space of $\mathcal C(I)$ generated by the indicatrix functions of the form $1_{[a,b]}$, $a,b\in I$. These are the step functions.

3. Define a linear form $\int : V \to \Bbb R$ by the formula $$\int 1_{[a,b]} = |b - a|$$ At this point, you have to check that this definition is well-founded by checking that if the finite sum $\sum_i \alpha_i 1_{[a_i,b_i]}$ is the zero function, then $\sum_i \alpha_i |b_i - a_i|$ is zero as well.

4. Check that $\int$ is $|I|$-lipschitzian, where $|I|$ is the length of $I$, that is to say $$\left|\int (f - g)\right| \leqslant |I|\| f - g\|_\infty$$

5. Remind this wonderful and easy result :
   Let $X$ and $Y$ be metric spaces, and assume that $Y$ is complete. For every $V$ subset of $X$ and $f : V\to Y$ uniformly continuous, then there exists a unique extension of $f$ to $\overline V$ which is uniformly continuous.

6. Apply this result with $\mathcal C(I)$, $\Bbb R$, $V$ and $\int$, to obtain a linear form $$ \int : \overline V \longrightarrow \Bbb R$$ This is the Riemann integral on $I$

7. Check that $\overline V$ contains all the continuous functions, and much more, and is complete. The functions in $\overline V$ are called ruled functions, or regulated functions. Note that you don't obtain every Riemann-integrable function this way (cf. Sam L. comment).

How ever, it should be noted that even if this approach simplify the construction in itself, it does not simplify the proofs about the riemann integral any further, and it is very important to have a concrete representation for this completion $\overline V$.

EDIT
I've been kindly asked by email to detail this answer, so here I am ! I don't know any book or other reference having this point of view, I'd be happy if somebody had a one.