Riemann integration in abstract spaces

Riemann integration in abstract spaces September 30, 2009

Posted by Akhil Mathew in analysis.
Tags: International Science and Engineering Fair, ISEF, Lebesgue integrals, open source triumphalism, Riemann integrals, self-embarrassment
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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!).

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Comments»

1. Julia - December 1, 2009: Hi Akhil,
Sorry to hear about the journal not being interested in the paper.
That happens to a lot of papers, of course… It’s a pity that the literature about the Riemann integral in more general spaces is not so obvious to find; there are papers our there, some sources in German an French. I wonder which one the referee mentioned to you. I actually came across your blog by chance when looking for a reference about the Riemann integral on compact metric spaces in English. Not sure what you could do to make it come up via google, ahead of a lot of other things (which are less related to the topic).
If you’re still interested in the subject, you may have a look into what
the class of R-integrable functions actually looks like, because there are some interesting examples out there. The classic theorem
says R-integrable is equivalent to mu-almost everywhere continuous,
and this also yields in more general spaces (e.g. loc. compact metric).
That result goes back to (at least) the 1950s (e.g. Bauer 1956 and earlier references there), but has been proved (and even published) again and again over the years, as late as 2005…
Good luck with your future work and studies.

Reply
Akhil Mathew - December 1, 2009: Thanks for the comment! Yes, I was interested in how much could work in this framework–and I will look into what you said about R-integrability and a.e. continuity.

I am copying the citations that the (anonymous) referee sent below, together with the comments:

Many authors assume that the measure is -additive on a semi-ring and continue to build the Lebesgue
integral. A. C. Zaanen: Introduction to the theory of Integration. North-Holland Publishing Co.,
Amsterdam, 1958

It is rarer to see a genuine theory of Riemann integral (for real-valued functions on semi-rings), since the
motivation (in the -additive case, which occurs in all basic examples) is unclear. Nevertheless, there are
papers that consider finitely additive measures in this setting or prove a statement that is valid for the
Riemann integral but not for the Lebesgue integral. For example
Strydom, B. C.: Abstract Riemann integration. Getal en figuur, 10 Van Gorcum & Co. N. V., Assen 1959;
Luxemburg, W. A. J.: The abstract Riemann integral and a theorem of G. Fichtenholz on equality of
repeated Riemann integrals. IA and IB. Nederl. Akad. Wetensch. Proc. Ser. A 64 (1961) 516-533, 534-545.

(2) Non-overlapping intervals appeared again in connection with generalized Riemann or Kurzweil-Henstock
integral. Here they make sense: these authors use finite tagged partitions and, already on the line, in can
happen that for all partitions satisfying their requirements (which are much stronger than -fine) some tags
are forced to be end points of their intervals. So the intervals have to be closed in order to contain their tags.
Of course, these authors are not interested in Riemann integral, but in the generalized one (which integrates
more functions than Lebesgue). An approach closest to the present paper is in
Rieˇcan, B.: On the Kurzweil integral in compact topological spaces. Rad. Mat. 2 (1986), no. 2, 151–163
Radovi Matematiˇcki may be rather difficult to find, but the definitions are repeated in
Boccuto A.; Rieˇcan, B.: A note on a Pettis-Kurzweil-Henstock type integral in Riesz spaces. Real Anal.
Exchange 28 (2002), no. 1, 153-162
There are many accessible texts on Kurzweil-Henstock integral, starting (on the real line) with
Henstock, R.: Theory of integration. Butterworths, London 1963.
The number of various generalized versions is too large to attempt even a slightly representative selection.
One of the first was
Henstock, R.: Definitions of Riemann type of the variational integrals. Proc. London Math. Soc. (3) 11
1961 402–418.

There have been some attempts to give a “constructive” definition of (Riemann) integral in abstract
spaces (in connection with computer science), for example
1
Edalat, A.: Domain theory and integration. Theoret. Comput. Sci. 151 (1995), no. 1, 163–193.
Edalat, A.; Negri, S: The generalized Riemann integral on locally compact spaces. Topology Appl. 89 (1998),
no. 1-2, 121–150.
Their definitions are rather abstract, but if one tries to understand them in a more standard way, one doesn’t
end too far from some ideas of the present paper. But the direction is closer to Lebesgue integration via
Riemann type procedure than to Riemann integration. In particular, no “intervals” are postulated.

For many authors, Riemann integral (in a topological situation) is defined as Lebesgue integral restricted
to functions that are bounded and continuous almost everywhere; or the goal is to show that the new
definition is equivalent to this one. [This is how one can understand the Riemann bit in (3).]

There is a large number of other relevant directions, from which I mention only
Ridder, J.: Die allgemeine Riemann-Integration in topologischen R¨aumen. A, B. Nederl. Akad. Wetensch.
Proc. Ser. A 71 (1968), 12-23; ibid. 71 1968 137–148.
Some information may be also found in corresponding articles in Handbook of measure theory. Vol. I, II.
North-Holland, Amsterdam, 2002. (Unfortunately, the Handbook is devoted mainly to other directions, and
so many of the developments relevant here have not been included.)

Reply
2. Julia - December 3, 2009: That is a remarkable number of references you got from the referee -
actually, it’s amazing! Interesting it doesn’t contain the even older ones (from the 1950s, see below), which contain the whole integral construction on very general spaces and some further results regarding the interaction between topology and measure/integral. (Well, they are inconvenient to read; not only are they in German, but also do they use those old fracture letters that not even Germans know how to pronounce anymore.) In particularly, they have that result I mentioned characterizing the class of R-integrable functions. There’s a paper about just this, for the (special) case of compact metric spaces is this (Beer, Int. J. Math. & Math. Sci, 1978, vol. 1). Strangely enough, it doesn’t cite any of the existing literature on this. But it’s self-contained and easy to read. Doob also summarizes this in his book on measure theory (VI, 20. on p.98).

Do you know this beautiful example for a R-integrable function?
x in [0,1]. If irrational, f(x)=0. If rational, represent x=p/q with
p, q non-neg. integers, no common divisor, and set f(x)=1/q.

\bibitem{Bau56}
H.~Bauer.
\newblock {{\:U}ber die Beziehungen einer abstrakten Theorie des Riemann-Integrals zur Theorie Radonscher Masse.}
\newblock {\em Math. Z.}, 65, 448–482, 1956.

\bibitem{Doob93}
J.L.~Doob.
\newblock {\em {Measure Theory.}}
\newblock Springer-Verlag, New York, Heidelberg, Berlin, 1993.

\bibitem{HauP55}
O.~Hauptmann and C.~Pauc.
\newblock {{\em Differential- und Integralrechnung, Band III, 2. Auflage.}}
\newblock G{\:o}schen Lehrb{\:u}cherei, 26, Berlin 1955.

\bibitem{Loo54}
L.H.~Loomis.
\newblock {\em {Linear functional and content.}}
\newblock {\it Amer. J. Math.}, 76, 68–82, 1954.

Best wishes,

Reply
Akhil Mathew - December 4, 2009: Yes, that’s a nice example–I think it’s in Rudin’s analysis book somewhere as an exercise (f is discontinuous at x iff x is rational, and bounded, so R-integrable). Or if not there, in some similar real-analysis text.

Thanks again for the references!

Reply

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