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also known as

the generalized Riemann integral, the Henstock integral,

the Kurzweil integral, the Henstock-Kurzweil integral,

the HK-integral, the Denjoy-Perron integral, etc.

Your comments are invited.

Still later, Riemann refined the definition that Cauchy had been using, and investigated the integrals of discontinuous functions. The Riemann integral is simpler to define than any of the other integrals discussed below, and it is the "standard" integral that we teach to undergraduate students. Still, it is more complicated than the antiderivative, and it seems to be beyond the understanding of many undergraduate students.

The theory of the Riemann integral was not fully satisfactory. Many important functions do not have a Riemann integral -- even after we extend the class of integrable functions slightly by allowing "improper" Riemann integrals. Moreover, even for integrable functions, it is difficult to prove good convergence theorems using only the tools ordinarily associated with Riemann integrals. A pointwise, bounded limit of Riemann integrable functions is not necessarily Riemann integrable. (For instance, since the rationals are enumerable, the characteristic function of the rationals can be represented as the pointwise limit of a sequence of characteristic functions of finite sets.)

In 1902, Henri Lebesgue devised a new approach to integration, overcoming many of the defects of the Riemann integral. Lebesgue's definition is appreciably more complicated, but Lebesgue's techniques yield better convergence theorems and, for the most part, more integrable functions. The Lebesgue integral has become the "standard" integral in our graduate courses in analysis.

The Lebesgue integral is strictly more general
than the proper Riemann integral -- i.e., it can integrate
a wider class of functions. However, in comparing the
*improper* Riemann integral with the Lebesgue integral, we find
that neither is strictly more general than the other.
Examples will be given later in this article.

Neither the improper Riemann integral nor the Lebesgue integral yielded a fully satisfactory construction of antiderivatives. Slightly more satisfactory answers -- i.e., more general notions of integral -- were given by Arnaud Denjoy (1912) and Oskar Perron (1914). Denjoy's and Perron's definitions turned out to be equivalent; both were very complicated.

Decades later, independently, Ralph Henstock (1955) and Jaroslav
Kurzweil (1957) found a much simpler formulation of the Denjoy-Perron
integral. In fact, the Henstock-Kurzweil formulation -- the **gauge
integral** -- is considerably simpler than the Lebesgue idea, and its
definition is only slightly different from the definition of the
Riemann integral. Consequently, interest in this integral has been
rising over the last few decades, and some mathematicians have even
advocated that we should teach the gauge integral either alongside or
in place of either the Riemann integral or the Lebesgue integral.
I will express my own opinions about that possibility, at the
bottom of this web page. (In January 1997 I circulated
a letter advocating the use of the gauge
integral. It was signed by Henstock, Kurzweil, and several
other leaders of the gauge integral movement. I distributed it
to several publishers of calculus textbooks. However, it apparently
hasn't had any effect.)

Henstock, Kurzweil, and other early researchers in this area were writing in rather specialized and technical terms for advanced audiences, so their ideas did not spread quickly at first. Here is a partial bibliography:

- McLeod's expository book,
*The Generalized Riemann Integral*(1980), was intended largely for*teachers*of integration theory. - Mawhin,
*Introduction à l'analyse*2nd edition (1981). Contains some material on the gauge integral. (I don't know whether that was also true of the first edition.) - Muldowney,
*General theory of integration in function spaces, including Wiener and Feynman integration*, 1987. - DePree and Swartz,
*Introduction to Real Analysis*, 1988, was intended for advanced undergraduate*students*. It includes two chapters of very readable material on the gauge integral. *Lanzhou Lectures on Henstock Integration*by Lee Peng-Yee, 1989, a rather clear exposition of research-level material.*The integrals of Lebesgue, Denjoy, Perron, and Henstock*by Russell Gordon, 1994. This book is a comprehensive introduction to the subject.- Mawhin,
*Analyse*, first edition (1992); 2nd edition (1997). - Bartle's article,
"Return to the Riemann integral," in the October 1996
*American Mathematical Monthly*, is brief and easy to read; probably it was read widely. - My book, published in October 1996, was intended for beginning graduate students; I hope it is helpful. It includes a chapter on the gauge integral, plus parts of several other chapters.
- A substantial number of articles in
*Real Analysis Exchange*are devoted to new discoveries concerning various aspects of the gauge integral. These articles make up perhaps 5 or 10 percent of the journal's content. This is*the*journal to read, for anyone who wants to do research in this area. Of course, most of these are research articles, not introductory articles; they will be unreadable to anyone who does not already know the basics of the subject. -- I might mention in particular that Gordon's article in*Real Analysis Exchange***22**(1) 1996/7, pp. 23-33, gives a historical overview of this subject in much more detail than the present web page. - Gordon's article in
*American Mathematical Monthly*, February 1998, discusses tagged partitions and gauges in interesting ways. - Robert G. Bartle and Donald R. Sherbert,
*Introduction to Real Analysis*, Third Edition, Wiley, 2000. Suitable for advanced undergraduates. Includes a chapter on the gauge integral. - Peng Yee Lee and Rudolf Vyborny,
*The Integral: An Easy Approach after Kurzweil and Henstock*, Cambridge University Press, 2000. A combined textbook/research monograph, suitable for advanced undergraduates or beginning graduate students. Very up-to-date -- for instance, it includes Bartle's recent theorem on negligible variation. - Jaroslav Kurzweil,
*Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces*. World Scientific Pub Co., 2000. This book is intended only for advanced specialists in integration theory, not for a general audience. It investigates topologies on the vector space of all gauge-integrable functions. - Robert Bartle,
*A Modern Theory of Integration*, Graduate Studies in Mathematics, AMS, 2001. Suitable for advanced undergraduates or beginning graduate students. - Charles Swartz,
*Introduction to Gauge Integrals*, World Scientific Pub Co., 2001. Suitable for advanced undergraduates.

Now a very slight modification yields the definition of the gauge integral. Again, there are several different, equivalent ways to define the gauge integral; we shall give only the formulation which emphasizes the similarity between the Riemann and gauge integrals.

The only change is in the use of delta. Instead of a positive constant, delta must be a positive-valued function. The function delta is called a

en español

The collection of numbers n, s_{i}, t_{i}
is called a **tagged division** (or a *tagged
partition*); the numbers s_{i} are called
the **tags**. The tagged division is called
**δ-fine** if it satisfies
the condition t_{i}-t_{i-1} <
δ(s_{i}) for all i.
Some of the books and papers on this subject use
a slightly different condition, e.g., that
s_{i}-δ(s_{i})
< t_{i-1} < t_{i} <
s_{i}+δ(s_{i}).
This affects the development of the theory only
superficially -- e.g., in some places we need to
replace an ε with either
2ε or
ε/2.

**Cousin's Lemma** states that for any gauge
δ, there *exists* at least
one δ-fine tagged division.
Cousin's Lemma is not obvious; that is
admittedly a drawback in the gauge theory.
The proof of Cousin's Lemma uses the
fact that the
real number system has the Least Upper Bound property.

There are several slightly different ways to prove Cousin's Lemma; here is a sketch of one of the proofs. Let S={x∈[a,b] : there exists a δ-fine tagged division on [a,x]}. Then S is nonempty, since a∈S. Also, S is bounded above by b. By the Least Upper Bound Property, the set S has a supremum, or least upper bound; let us denote that number by σ. Then σ∈[a,b]. If σ < b, use the fact that δ(σ) > 0 to arrive at a contradiction. If σ = b, use the fact that δ(b) > 0 to prove that the supremum is actually a maximum -- i.e., that b is a member of S.

The set of all gauge-integrable functions from [a,b]
to **R** is a vector space -- i.e.,
any constant multiple of such a function or any
sum of two such functions is another such function.
That vector space is often known as the
**Denjoy space**, because of some work Denjoy
did in studying it.

We have only considered the simplest possible versions of the definitions, but we remark that some generalizations are possible.

- With slightly more complicated definitions, we could allow
plus and minus infinity as possible values
for the integrals. (Just equip the extended real line
with its usual convergence structure, and use
convergences instead of epsilons. This variant is used in the appendix
to Anatole Beck's book,
*Continuous Flows in the Plane*.) - Or, instead of working with real-valued functions f, we could work with functions f taking values in any Banach space X; most of the main properties generalize easily to this setting. (See my book, for instance.)
- We could also replace [a,b] with other domains, but that is harder; it is discussed below.
- Henstock and Kurzweil have also considered replacing the
expression f(s
_{i})(t_{i}–t_{i–1}) with a more general expression g(s_{i}, [t_{i–1},t_{i}]), where g is a function of a real variable and an interval variable; but the results in that case are more complicated and will not be considered here.

However, the fact that we're using properties of intervals means that the gauge integral does not generalize readily to settings other than intervals. It can be generalized to functions defined on the whole real line, or to functions defined on finite-dimensional spaces, but its definition and theory then become slightly more complicated. It apparently can also be generalized to infinite dimensional spaces, and more abstract setttings, but I'm not very familiar with this theory, and I believe it's a lot more complicated. Some of Henstock's writing deals with this theory. I believe that Muldowney's book gives a more elementary treatment of this theory, but I have to confess that I'm not very familiar with the theory or with his book.

Contrast that with the Lebesgue approach, which requires far more prerequisite equipment (sigma-algebras, measures, measurable sets, measurable functions) but then generalizes without any difficulty at all to settings entirely unrelated to intervals. Thus, the Lebesgue approach may be preferable if one wants to study stochastic processes such as Brownian motion. (Brownian motion can be explained in terms of Wiener measure, a measure on an infinite-dimensional space of continuous functions.)

Here is an example of the concreteness of the gauge integral: Any
introductory course on measure and integration should include a
construction of Lebesgue measure, but how should that be done? The
usual constructions are rather complicated and abstract; they have
little to do with the *uses* of Lebesgue measure. But it is
easy to define the gauge integral, and after we've defined it we can
proceed this way: Define

As indicated by the Venn diagram above, not every Lebesgue integral can be viewed as a Riemann integral, or even as an improper Riemann integral. For a simple example, consider the characteristic function of the rationals.

As indicated by the Venn diagram above, not every improper Riemann
integral is a Lebesgue integral. For an example, let f(t) =
t^{2} cos(1/t^{2}) with f(0)=0. Then
_{0}^{1}|f '(t)|dt
is infinite, and the Lebesgue integral
_{0}^{1}f '(t)dt
does not exist, but the improper Riemann integral
_{0}^{T}f '(t)dt exists (for
any T) and equals f(T).

The examples above are typical. If g(t) is any gauge
integrable function on [a,b], then g(t) is Lebesgue
measurable. Hence the Lebesgue integral _{a}^{b}|g(t)|dt exists, if we
permit infinity as a possible value. The Lebesgue
integral _{a}^{b}g(t)dt exists (with
its usual definition) if and only if _{a}^{b}|g(t)|dt is
finite. For any nonnegative function, the Lebesgue and
gauge integrals are the same.

This analogy may be helpful: The gauge integrable functions
are like convergent series; then the Lebesgue integrable
functions are like *absolutely* convergent series.
The absolutely convergent series are easier to work with,
and yield a tidier theory. The convergent series are
more general, but only occasionally do we need to work with
a series that is conditionally convergent.

Rapidly oscillating functions such as the derivative of t^{2}
cos(1/t^{2})
are the only ones added to our collection of integrable functions when
we switch from the Lebesgue integral to the gauge integral. These
functions are more often used for pathological
counterexamples than for positive applications. Thus the
greater generality of the gauge theory does not make it
much more applicable. (A possible exception is in the
theory of differential equations; there seem to be some
important new applications there.) For the most part,
the big advantage of the gauge integral is the new
insight that it yields into the Lebesgue integral.

Here is another drawback to the gauge theory: The Lebesgue theory
yields the Banach space L^{1}[a,b] of all Lebesgue-integrable
functions. It's a very nice space, with a nice metric and nice
convergence theorems. The gauge theory yields a slightly larger vector
space, which we might call G[a,b]. But the metric and the convergence
theorems for this space are much less nice. That's not surprising:
this space contains not only the nice functions of L^{1}[a,b],
but also some additional functions that are rather nasty. If we don't
really need to work with those nasty functions, we might prefer to
stick to the setting of L^{1}[a,b], which has nicer worktools
(i.e., the metric, the convergence theorems, etc.).

Result (a), or one like it, is what we teach our graduate students, under the stronger assumption that f is Lebesgue integrable. Result (b), or one like it, is what we teach our undergraduates -- usually under the assumption that f is continuous throughout

First Fundamental Theorem of Calculus (derivatives of integrals).Let f be a real-valued, gauge integrable function on [a,b], and let F(t) =_{a}^{t}f(s)ds. Then:

(a)- the derivative F '(t) exists and equals f(t), for almost every t in [a,b].

(b)- the derivative F '(t) exists and equals f(t), at each t where f is continuous.

We teach our undergraduates a theorem like this, but it usually has the additional hypothesis that the Riemann integral

Second Fundamental Theorem of Calculus (integrals ofderivatives).Let F be a real-valued, differentiable function on [a,b]. Then the gauge integral_{a}^{b}F '(t)dt exists and equals F(b)–F(a).

Actually, the hypotheses can be made even weaker: We can
allow
F to be nondifferentiable at countably many points. More
precisely: Assume F is continuous, and that G is some
function
defined at every point of [a,b], and that F'(t) exists
and
equals G(t) for all but at most countably many values
of t. Then the gauge integral
_{a}^{b} G(t)dt
exists
and equals F(b)–F(a).

Thus, any "improper gauge integral" also exists as a proper one.

(*Clarification*: The function f must be defined at
*every point* of [a,b]. In the case of
f(t)=1/√t on the
interval (0,1], we also assign some
value to f(0). What value we assign doesn't really
matter, just so long as f(0) is some particular number.)

But uniform convergence is a very strong assumption, so we usually teach our graduate students a result like this:

Suppose (f _{n}) is a sequence of Riemann integrable functions on [a,b], converging uniformly to a limit f. Then f and |f_{n}–f| are Riemann integrable, and as well.

We usually teach this to our graduate students under the additional assumption that the functions e and g are Lebesgue integrable; then the other functions are Lebesgue integrable too. However, the version of the theorem presented above can be stated (without proof) in undergraduate courses -- perhaps even in a freshman calculus course.

Dominated Convergence Theorem.Suppose (f_{n}) is a sequence of gauge integrable functions, converging pointwise to a limit f. Also suppose that e(t)<f_{n}(t)<g(t) for all n and t, where e and g are some gauge-integrable functions. Then f and |f_{n}–f| are gauge integrable, and as well.

There are other convergence theorems for gauge integrals, involving
hypotheses weaker than the domination condition e(t)
__<__ f_{n}(t) __<__ g(t),
but those theorems are generally much more complicated. Research is
still being done in that area.

Here are my opinions (substantially revised 11/16/00). The role of the gauge integral in our teaching must be viewed in terms of the way that we presently teach analysis. Here in the USA, we usually teach analysis on three levels:

- The freshman calculus course, which uses the Riemann integral and omits most proofs.
- An advanced undergraduate course, typically using a textbook titled "Introduction to Real Analysis" or something like that. This course generally teaches how to do analysis proofs; it includes proofs of the results that were stated without proof in calculus.
- A beginning graduate course, which generally covers the Lebesgue integral.

Level (2), on the other hand, would be an excellent place to introduce the gauge integral. The conventional course at the advanced undergraduate level concentrates on the use of tools such as epsilons and deltas, convergent sequences, limsups and liminfs, etc. Those are the same tools used in the theory of the gauge integral. Thus, adding that integral to the course would only involve slight modifications in the course. The book by DePree and Swartz is particularly well suited for this purpose. Bartle and Sherbert, 3rd edition, is also applicable to this purpose, though it devotes slightly fewer pages to the gauge integral.

Introducing the gauge integral into level (3), the graduate course, might also be a good idea, but it would involve greater complication. The graduate course has its share of epsilons and limsups, but its most basic tools are topologies and sigma-algebras, a very different sort of thinking. The gauge integral does not generalize readily to abstract settings like abstract sigma-algebras. (It can be generalized to a setting something like that, but it then loses much of its wonderful simplicity.) In its simplest form, it is more like the Lebesgue integral with respect to Lebesgue measure, not other measures. Of course, Lebesgue measure is the most important measure. (Perhaps we should deemphasize the other measures in the first graduate course on integration theory, in order to make the gauge integral fit into the course more readily. Other measures can be developed in greater detail in a later course.)

I think that the two theories -- Lebesgue and gauge -- both are valuable, and both deserve to be covered in the graduate course. I'm not yet sure what is the best way to organize such a course. We can develop one of the two theories, and then use its results as tools in developing the other theory. This is the approach taken by Gordon's book (which develops the Lebesgue theory first) and by Bartle's "MTI" book and the Lee/Vyborny book (both of which develop the gauge theory first).

Of course, there are some obvious
examples -- e.g., if we permit the function to be unbounded
or if we permit its domain to be unbounded. But what
about **a bounded function on a bounded interval**?
Within this class of functions, it is hard to give
an example of a function that is not integrable.
It was already hard enough with the Riemann integral -- for that
integral we had to use rather bizarre functions, such as the
characteristic function of the rationals. Now, when we turn to the
gauge integral or the Lebesgue integral, more
functions are integrable, and so it is even harder
to produce examples of non-integrable functions.

Actually, the problem is the same for gauge integrals or Lebesgue integrals. Indeed, when f is a bounded real-valued function on a bounded interval, then these three conditions are equivalent:

- f is gauge integrable;
- f is Lebesgue integrable;
- f is Lebesgue measurable.

"every bounded function on a bounded interval is gauge integrable."Indeed, there do

Every introductory textbook on Lebesgue integrals includes a short
proof (due to Vitali) of the existence of a nonmeasurable set;
the characteristic function of that set is then a
nonmeasurable function. But
that proof, like every known proof of that theorem, is
*nonconstructive* -- it uses the Axiom
of Choice to prove the
existence of the nonmeasurable function without actually "finding" the
function or describing it explicitly. Vitali's proof is one of
the most elementary uses of the Axiom of Choice, and perhaps
it makes a good introduction to the Axiom of Choice; it could
be included in an appendix in a book intended for some advanced
undergraduate students. But I think this is conceptually
way beyond the
grasp of freshman calculus students. All we can do is tell these
students: "Yes, there does exist a non-integrable, bounded function on
a bounded interval, but describing it to you would require a great deal
of higher math, far beyond the scope of this freshman calculus course."

You might notice that, in the last paragraph, I said that every
*known* proof requires the Axiom of Choice.
Is it possible that we may someday find another
proof that is constructive? i.e., that we may find an *explicit example*
of an unmeasurable function? Probably not. This is not just a personal
opinion -- it has some supporting evidence, though the evidence is rather
technical. Solovay proved in 1970
that the nonexistence of a nonmeasurable function is consistent with ZF
set theory (i.e., conventional set theory minus the Axiom of Choice),
provided that there exists an inaccessible cardinal. The existence of
an inaccessible cardinal is *empirically* consistent with conventional
set theory -- i.e., after many decades, no one has yet found
a contradiction, so Solovay's model seems to be reliable.
If you're interested, these topics are discussed a little more in
my book, and more references are given there. --
My, we certainly seem to have gotten into some deep math, while
discussing what to teach to freshmen!

*Give a descriptive definition of the Riemann integral.*Saks mentioned in his book that you can approach an integral descriptively (like Newton -- e.g., the integral is some sort of antiderivative) or constructively (like Riemann -- e.g., the integral is some sort of limit of approximating sums). Many results we have proved for the gauge integral are not applicable to the Riemann integral, and the above-mentioned question is one of them.- Lee Peng-Yee has given a Riemann-type definition
of the Ito integral using the Henstock
method. It definitely includes the Ito integral but
Lee Peng-Yee has not yet been able to determine
whether it also includes anything else -- i.e., is
it equivalent to the Ito integral, or is it more general?
*Is there a natural topology for the Denjoy space?*The sup norm gives us a natural topology on C[0,1], and the norm || · ||_{1}gives us a natural topology for L^{1}[0,1]. Is there a topology that is somehow natural for the Denjoy space (i.e., the space of all gauge-integrable functions)? We do know many convergence theorems for the gauge integral, as mentioned earlier on this web page, but so far they have not provided a good answer to this question about topologies. Several topologies for the Denjoy space have been proposed in the literature, but none of them seems to have quite enough properties for us to call it a "natural" topology. Some properties that would be pleasant would be: metrizable; complete or barrelled or ultrabarreled; locally convex; and corresponding in some natural way to some of the best known convergence theorems about gauge integrals. We may be able to get some of those properties, but apparently not all of them; that is made evident by discoveries in Kurzweil's recent book on this subject.