Riemann integration in abstract spaces

I’ve been busy as of late with college applications and a science competition. But now I have a bit more time, so I shall try to resume posting.

Anyway, speaking of science competitions, I participated in the Intel International Science and Engineering Fair in 2007 with a self-guided project. The bulk of it dealt with Riemann integration in abstract spaces and the potential for generalizing certain constructions in analysis to this setting.

After the competition, I tried submitting a condensed version of the material to a mathematical journal, which concluded that the work did not merit publication, but may have had some interest: the method, while contained in more general approaches, seemed to have not been taken in the literature. (Unfortunately, I was unaware of the literature.)

The paper I submitted is here.

Nevertheless, since this is not a professional blog, I thought this might be an appropriate setting to post the paper and briefly discuss it, so I will try and see how this goes.

The Riemann-Darboux Integral

As is well-known, the Riemann-Darboux integral of a function ${f: [a,b] \rightarrow \mathbb{R}}$ is defined as follows. One splits ${[a,b] = \bigcup_i I_i}$ for ${I_i \subset [a,b]}$ a collection of intervals such that ${I_i \cap Int(I_j) = \emptyset}$ for ${i \neq j}$ ; this is a partition ${P}$ . One defines the upper and lower sums

$\displaystyle U(P,f) := \sum_i \sup_{x \in I_i} f(x) \ length(I_i), \quad L(P,f) := \sum_i \inf_{x \in I_i} f(x) length(I_i)$

and defines ${f}$ to be integrable if ${\inf_P U(P,f) = \sup_P L(P,f)}$ , and calls the common value the integral.

Just as the Lebesgue integral can be defined on abstract spaces (and is perhaps most naturally done this way), we can abstract the notion of “intervals” and “length” to a compact metric space ${X}$ . So, on ${X}$ a family of intervals—which may be thought of as some weak form of a ${\sigma}$ -algebra—is defined as a subset ${\mathfrak{I} \subset P(X)}$ that satisfy:

${X \in \mathfrak{I}}$ ,
If ${A, B \in \mathfrak{I}}$ , then ${A \cap B \in \mathfrak{I}}$ ,
If ${\delta > 0}$ , there exist ${A_1, A_2, \dotsc A_n \in \mathfrak{I}}$ such that ${X = \bigcup_{i=1}^n A_i}$ , ${A_i \cap \mathrm{Int} {A_j}= \emptyset}$ for ${i \neq j}$ , and ${\mathrm{diam} (A_i)<\delta}$ for each ${i}$ .

The last condition allows for small coverings—this is necessary to prove that a continuous function is integrable. Similarly, to generalize the notion of a length function, we choose some ${\mu: \mathfrak{I} \rightarrow \mathbb{R}_{\geq 0}}$ with

${\mu(A) \geq 0}$ for all ${A \in \mathfrak{I}}$ ,
If ${A \in \mathfrak{I}}$ has empty interior, ${\mu(A)=0}$ ,
If ${A_1, A_2 \dotsc A_m}$ satisfy ${A_i \cap \mathrm{Int} {A_j} = \emptyset}$ for ${i \neq j}$ and ${A= \bigcup_n A_n \in \mathfrak{I}}$ , then ${\mu(A) = \sum_n \mu(A_n)}$ .

Anyway, an example of this is, of course, closed subintervals of a compact interval ${I \subset \mathbb{R}^n}$ , with the length function as the volume. In ${\mathbb{R}}$ , if we have an interval ${I}$ with a nondecreasing function ${\alpha: I \rightarrow \mathbb{R}_{\geq 0}}$ , then we can pick subsets ${[c,d) \subset I}$ as a family of intervals and ${\mu_{\alpha}( [c,d)) := \alpha(d) - \alpha(c)}$ as a length function.

Ok, now for integration. Given a partition ${P}$ of ${X}$ , i.e. a finite union

$\displaystyle X = \bigcup_i I_i,$

where ${I_i \in \mathfrak{I}}$ and ${I_i \cap Int(I_j) = \emptyset}$ when ${i \neq j}$ , we define the upper and lower sums similarly as in the real case, set the upper integral to be the inf of the upper sums and the lower integral to be the sup of the lower sums. Call ${f}$ integrable when the upper and lower integrals coincide.

Our examples yield, respectively, the Riemann integral in ${\mathbb{R}^n}$ and the Stieltjes (i.e. Darboux-Stieltjes) integral.

Theorem 1 A continuous function is integrable.

Indeed, ${X}$ is compact so we have uniform continuity. The proof is essentially the same as the standard one.

There are lots of standard properties here, i.e. linearity, monotonicity, etc. But this is an exercise in repeating standard textbook real-analysis proofs, so let’s move on.

Changing Variables

It turns out that we can change variables in this context too. We don’t, of course, have a nice way to differentiate functions. But we can differentiate length functions, and this is done in a manner reminescent of differentiating measures in Euclidean space with respect to Lebesgue measure. I’ll sketch the ideas here (though in the paper I take a bit more generality).

So, if we have two length functions ${\mu_1, \mu_2}$ on the same space ${X}$ with the system of intervals ${\mathfrak{I}}$ , we say that ${\mu_2}$ is differentiable with respect to ${\mu_1}$ at ${x \in X}$ if we can choose ${A \in \mathbb{R}}$ such that for each ${\epsilon>0}$ , there is a ${\delta>0}$ such that ${x \in J}$ , ${diam(J) <\delta}$ imply

$\displaystyle \left|{ \mu_2(J) - A \mu_1(J) }\right| \leq \epsilon \mu_1(J) .$

This isn’t necessarily unique, but it will be if ${\mu_1(J) \neq 0}$ for ${J}$ containing ${x}$ and of nonempty interior.

If ${A}$ is everywhere defined, it is a function ${\frac{d \mu_1}{d \mu_2}}$ on ${X}$ . In the continuous case, there is a “mean value theorem.”

Theorem 2 If ${f(x) = \frac{d \mu_2}{d \mu_1}(x)}$ is continuous and ${\mu_1}$ is nonvanishing at intervals of nonempty interior, then there is ${\xi \in X}$ with ${f(\xi) = \frac{ \mu_2(X)}{\mu_1(X)}}$ .

Instead of writing out the proof, I’d like to sketch how it reduces for the case of ${X}$ an interval ${[a,b]}$ on the real line and ${\mu_1}$ is the usual length, when it is a special case of the usual mean value theorem (and an elementary exercise). So we have a ${\mu_2}$ , i.e. a nondecreasing and continuously differentiable ${g: [a,b] \rightarrow \mathbb{R}}$ . We must prove that there is a ${\xi \in [a,b]}$ with

$\displaystyle \frac{g(b)-g(a)}{b-a} = g'(\xi);$

we cannot use the usual maxima proof because maxima and minima don’t make any sense in the context we’re trying to generalize to. So suppose that ${length([a,b])(g'(\xi) + \eta) < g(b)-g(a)}$ for all ${\xi \in [a,b]}$ . Then the same must hold by replacing ${[a,b]}$ by one of the subintervals ${\left[a, \frac{a+b}{2} \right], \left[ \frac{a+b}{2}, b \right]}$ , as is easily checked. Inductively keep bisecting in this manner to get a sequence of nested intervals ${I_i = [a_i,b_i]}$ with

$\displaystyle length([a_i,b_i])(g'(\xi) + \eta) < g(b_i)-g(a_i), \quad \xi \in I_i;$

the intervals converge to some point ${x}$ with ${g'(x) + \eta < g'(x)}$ , contradiction. This proof is the one I generalize in the paper.

With it, there is a change-of-variables formula.

Connection with the Lebesgue integral

As one might expect, it is possible to construct a measure from these “length functions” under suitable conditions that extends this generalized Riemann-Darboux integral in the same way that the Lebesgue integral in Euclidean space extends the usual Riemann integral. The machinery I invoke to get the measure from the length function is the Daniell integral. With it, I show that the derivatives above are just Radon-Nikodym derivatives—indeed, this is basically a corollary of the change-of-variables formula.

Anyway, blogging this was a reminder of how much real analysis has already evaporated since I wrote this. But I am hoping that this paper may be of some interest to passers-by on this blog, if only as a review of analysis (as it was for me!).

Climbing Mount Bourbaki