
Reviewed by J. Alan Alewine and Eric Schechter.
A simple definition.
Riemann's integral of 1867 can be summarized as

The values of f are used less directly in Lebesgue's integral (1902), which can be described as lim_{n®¥} ò_{a}^{b} g_{n}(t)dt. The approximating functions g_{n} must be chosen carefully, using deep, abstract notions of measure theory. Simpler definitions are possiblefor example, functional analysts might consider the metric completion of C[0,1] using the L^{1} normbut such a definition does not give us easy access to the Lebesgue integral's simple and powerful properties such as the Monotone Convergence Theorem. We generally think in terms of those simple properties, rather than the various complicated definitions, when we actually use the Lebesgue integral.
The KH integral (also known as the gauge integral, the
generalized Riemann integral, etc.)
was discovered or invented
independently by Kurzweil and Henstock
in the 1950's; it has attracted growing interest in recent
years. It offers the best of both worlds: a powerful
integral with a simple definition. In fact, its definition
is nearly identical to that of the Riemann integral, as
we now show. For any
tagged partition

We emphasize that the function d(t), though positive at each t, need not be bounded below by a positive constant. In applications we generally take d(t) to be extra small at locations t where f behaves erratically, so that those locations have less effect on the summation åf(t_{i})(t_{i}t_{i1}). Think of approximating the region under a curve as a union of thin rectangles, as in a calculus course: The Riemann integral simply requires that all the rectangles be narrower than a certain constant width, but the KH integral uses more sophisticated spacing.for each constant e > 0 there exists a constant d > 0 (respectively, a function d:[a,b]® (0,+¥)) such that whenever P is a dfine tagged partition of [a,b], then vf(P) < e.
For example, let 1_{Q} denote the
characteristic function of the
rationals. This function is discontinuous everywhere; it
is a standard example of a bounded function that is not
Riemann integrable on [0,1]. Nevertheless,
the KH integral ò_{0}^{1} 1_{Q}(t)dt exists and equals zero.
Indeed, let
p_{1},p_{2},p_{3},¼ be any enumeration of the rationals, and let

For a second example, consider the function f(t) equal to
t^{1}sin(t^{2}) on (0,1], and vanishing at t=0.
This function is neither Riemann nor Lebesgue integrable,
but it is KH integrable; that can be shown using

A more general integral.
Generalizing the constant d to a
function d(t) obviously
yields a wider class of integrands, but it is surprising
just how much wider. It turns out that



Kurzweil was led to Riemannlike integrals by his
investigations of differential equations
u¢(t)=f(t,u(t)). To see the
connection between the two topics, note that
some initial value problems can be restated as integral
equations:

The KH integral brings wider applicability to differential equations, to Fourier analysis (mentioned later in this review), and to some other branches of analysis, because the KH integral can integrate more functions. However, t^{1}sin(t^{2}) is typical of the new functions: They are erratic, more often cited for pathological counterexamples than for useful applications. Thus, the chief benefit of the KH theory may not be its wider applicability, but rather its concrete and elementary formulations of ideas that we already know in the Lebesgue theory. For example, a set S Í [a,b] is Lebesgue measurable if and only if its characteristic function 1_{S} is KHintegrable, in which case its Lebesgue measure is ò_{a}^{b} 1_{S}(t)dt.
For brevity, we have defined the KH integral only on a
compact interval, but the basic ideas extend easily to
bounded or unbounded regions in finitedimensional
Euclidean space, and to measures other than Lebesgue
measure. For example, it is shown in
Theorem 4.1.1 of [12] or Theorem 24.35 of
[13] that the measures m on the Borel subsets of
an interval [a,b] can be expressed as KHStieltjes
integrals:

The preceding characterization of measures makes good use of the special properties of intervals in R, admittedly a rather special setting. The KH integral can be extended to a more abstract and general setting of "division spaces'' (see, for example, [9]), and the resulting theory is applied to the Wiener and Feynman integrals in [11], but this theory is more complicated. Still, it is a continuation of the ideas of the Riemann integral; we do not have to start over with a whole new approach involving salgebras. This may make the KH approach attractive to scientists and engineers.
Teaching the KH integral. Where, if at all, does the KH integral belong in our standard analysis curriculum? Bartle [1] suggests that it could replace the Lebesgue integral, while Gordon [8] says that it should not. Perhaps the difference in their opinions reflects different audiences. For example, the KH approach permits us to avoid, or at least postpone, the notion of salgebras. Such abstract notions are insightful and valuable for a mathematically advanced audience, but may be less accessible for undergraduates or for scientists and engineers.
At many American universities today, the standard analysis curriculum is in three stages:
(1) Freshman calculus. The KH integral simplifies and strengthens some classical results. For example, one half of the Fundamental Theorem of Calculus says
If G:[a,b]® R is differentiable [and G¢ is continuous], then G¢ is integrable and ò_{a}^{b} G¢(t)dt = G(b)G(a).The continuity assumption, or some other assumption like it, is needed for Riemann integrability; that assumption can be omitted entirely if we use the KH integral.
Another improvement on calculus is Hake's Theorem (Theorem 2.8.3 in the LeeVýborný book):
In effect, this says that we do not need to define an "improper'' KH integral, analogous to the improper Riemann integral of calculus; any improper KH integral is also a proper KH integral. (Hake's Theorem is not valid for Lebesgue integrals: lim_{r¯0} ò_{r}^{1} t^{1}sin(t^{2})dt = 0.312¼, but ò_{0}^{1} t^{1}sin(t^{2})dt does not exist as a Lebesgue integral.)The KH integral ò_{a}^{b} f exists if and only if lim_{r¯a} ò_{r}^{b} f exists, in which case they are equal.
For the sake of improvements such as these, new calculus teachers might be tempted to introduce the KH integral, but experienced teachers may be less optimistic about their students' abilities. Most of our calculus students can learn computations, but lack the mathematical maturity for proofs. For them, the Fundamental Theorem of Calculus is simply the equation ò_{a}^{b}G ¢(t)dt = G(b)G(a), and discussions about the integrability of 1_{Q} are just gibberish. The value of the calculus course to these students does not lie chiefly in its proofs.
(2) The advanced undergraduate course. A general goal of this course is to enable students to understand analysis proofs. That is accomplished more specifically by practicing techniques of ed, convergent sequences, limsups, and the like. As it happens, those techniques are also the main tools in the KH theory. Thus, adding the KH integral to the advanced undergraduate course would require only small alterations in that course.
At least two textbooks are already available that support such a course: DePree and Swartz [5] and the third edition of Bartle and Sherbert [4]. Each of these books covers the material of a conventional undergraduate analysis course, but then adds a chapter on the KH integral. The additional chapter appears late enough in the book so that it is not crucial; thus teachers who are hesitant about the KH integral can adopt it at their own pace.
(3) The graduate course on Lebesgue integration. This
course traditionally
commits a large amount of time to plowing through the
terminology and lemmas of set theory,
Nevertheless, the results of the two theories are closely related, as we noted in (A) and (B). Consequently, results from either theory can be used as tools in the development of the other theory. Gordon's book [7] develops the Lebesgue theory first, and then uses some of its results in developing the KH theory. The first few chapters of this book would fit the traditional graduate course with few alterations; the book lies somewhere between textbook and research monograph.
The book under review. The book of Lee and Výborný goes in the other direction, developing the KH theory first and then using it to develop the Lebesgue theory. It is written to be used on several different levels, as explained in its preface: Chapters 13 and 67 might replace the standard courses that we have called "Stage 2'' and "Stage 3''. Even Chapter 1, on the Riemann integral, can be read on different levels, according as one includes or omits the material marked "optional''. Chapters 45 are more advanced reading intended for specialists in integration theory.
The book is rich in examples and applications. We were fascinated by Example 1.4.5, a construction over a page long producing an everywhere differentiable function whose derivative is bounded but not Riemann integrable. The applications include things such as Corollary 7.5.3, which states that the Fourier series of a KH integrable function is Abel summable almost everywhere to that function. Also included are some of the latest discoveries. For instance, in 1993 Výborný formulated the notion of "negligible variation'', which Bartle used in 1997 to characterize indefinite KHintegrals; that characterization is Theorem 3.9.1.
The book covers not only the KH integral but also an assortment of "other'' integralsnotably, the Denjoy integral (1912), the Perron integral (1914), and the SL integral (formulated by Lee and Výborný in 1993 based on the strong Lusin condition). Actually, these "other'' integrals all turn out to be equivalent to the KH integralthat is, they yield the same classes of integrable functions and the same numerical values for the integrals. Though the KH definition is already quite enough for beginners, the alternate definitions yield additional insights that may be helpful in certain kinds of research. For example, the value of ò_{a}^{b} f(t)dt is not affected if we alter f on a set of measure zero; that fact follows only indirectly from the KH definition, but very directly from the Denjoy or SL definition. Thus either of those definitions, and its associated ideas and techniques, might be useful in investigations that involve discarding null sets.
One more topic that deserves mention is that of convergence theoremsthat is, sufficient conditions for ò_{a}^{b} f_{n} ® ò_{a}^{b} f. The Dominated Convergence Theorem for Lebesgue integrals is a special case of the Vitali Convergence Theorem, which can be generalized to the KH setting as follows (Theorem 3.7.5):
Equiintegrability Theorem. Suppose (f_{n}) is a sequence of KH integrable functions on [a,b], convergent pointwise to some function f. Suppose that (f_{n}) is KH equiintegrable, in the sense thatThat theorem is probably general enough for a first course on integration, and for anyone except a specialist in integration theory, but some deeper results with weaker but more complicated hypotheses are included in the advanced chapters.
for each e > 0 there exists a function d:[a,b]®(0,+¥), such that whenever P is a dfine tagged partition, then sup_{n}f_{n}(P)ò_{a}^{b}f_{n} < e.Then f is KH integrable and ò_{a}^{b}f_{n}®ò_{a}^{b}f.
The theory of the KH integral has not yet settled down to a classical formulation, but already it is worthy of a place in our standard curriculum. The book of Lee and Výborný serves well as an introduction and reference for anyone interested in this topic. Other good sources are Gordon [7], which covers much of the same material from a somewhat different perspective, and the forthcoming book of Bartle [3].
Vanderbilt University, Nashville, TN 37240
aalewine@math.vanderbilt.edu
schectex@math.vanderbilt.edu