##
**Unified integration.**
*(English)*
Zbl 0551.28001

In this text Professor McShane introduces and studies with meticulous care a slight variant of a Riemann-type integral which was introduced originally by J. Kurzweil. This integral - called a gauge integral by Professor McShane - turns out to be equivalent to the Lebesgue integral and is conceptually quite similar to the Riemann integral. To briefly describe the gauge integral notice that the definition of the Riemann integral for a function f: [a,b]\(\to {\mathbb{R}}\) is equivalent to the following: there exists a real number A such that for every \(\epsilon >0\) there exists \(\delta >0\) such that if \(p=\{a=x_ 0<x_ 1<...<x_ n=b\}\) is a partition of [a,b] and \(t_ j\in [x_{j-1},x_ j]\) satisfies \([x_{j-1},x_ j]\subseteq [t_ j-\delta,t_ j+\delta],\) then \(| \sum^{n}_{j=1}f(t_ j)(x_ j-x_{j-1})-A| <\epsilon.\) As Borel noted the division of the interval is entirely independent of properties of the function f. The integral of Kurzweil is realized by replacing the existence of the positive constant \(\delta\) in the definition above by the existence of a positive function \(\delta\) on [a,b]; i.e., in this case it is required that \([x_{j-1},x_ j]\subseteq [t_ j-\delta (t_ j),\quad t_ j+\delta (t_ j)].\) The function \(\delta\)- called a gauge - can obviously be adjusted to take care of oscillations or other poor behaviour of the function f in certain regions, and the resulting integral of Kurzweil turns out to be equivalent to the Perron integral. Prof. McShane uses a slight variant of the Kurzweil integral where he drops the requirement that \(t_ j\in [x_{j-1},x_ j].\) The resulting integral then turns out to be equivalent to the classical Lebesgue integral, and it is this integral which is studied in the text.

Prof. McShane asks the following question: ”Can we take advantage of the close resemblance to Riemann integration to produce a unified theory that can be taught to students who have no intention of becoming research mathematicians, with just about the same level of difficulty as is encountered in ordinary courses, and that can also go from beginning calculus to the graduate level without ever abandoning earlier work and starting again (as usually now happens when Lebesgue integration is met)?” This text is an attempt to answer this question in the affirmative, and, in this reviewer’s opinion, it is a resounding success.

In Chapters 1 and 2, the gauge integral in one-dimension is introduced and its basic properties, including the Monotone and Dominated Convergence Theorems, are dicussed in detail. Chapter 4 discusses gauge integration in n-dimensions and includes versions of the Fubini and Tonelli theorems. Prof. McShane’s presentation of the gauge integral is painstakingly thorough and clear; he assumes minimal background on the part of the reader and presents proofs in complete detail. The presentation is clearly aimed at a reader with minimal background and preparation.

There is a very extensive list of applications presented in the text. Chapter 3 contains applications of the gauge integral to ordinary differential equations (its birth place) and probability. Chapter 5 contains discussions of both line and surface integrals. Chapter 6 contains applications to various topics in harmonic analysis; for example, \(L^ 1\) and \(L^ 2\), Hilbert spaces, orthogonal expansions and Fourier transforms. All of these discussions are very complete and thorough. Finally, Chapter 7 contains a dicussion of the Lebesgue integral with respect to an abstract measure. It is shown that this traditional approach to the Lebesgue integral is equivalent to the gauge integral in the case of Euclidean spaces. A number of additional topics, such as the Radon-Nikodym Theorem and Brownian motion, are also discussed.

Prof. McShane indicates his desire that the gauge integral will eventually become widely accepted and find its way into introductory analysis textbooks. While this book probably contains far too much material on integration and does not contain many other topics which are normally discussed in an introductory analysis course, it certainly does contain material from which a presentation of the gauge integral at an introductory level could be based. Hopefully, the mathematical community will accept Prof. McShane’s challenge and begin to incorporate a presentation of the gauge integral into introductory courses in real analysis.

{Reviewer’s remark: Several books have recently appeared which contain discussions of gauge-type integrals: ”The Generalized Riemann Integral” (1980; Zbl 0486.26005) by R. M. McLeod, ”Introduction à l’analyse” (1979; Zbl 0444.26002) by J. Mawhin, and ”Nichtabsolut konvergente Integrale” (1980; Zbl 0441.28001) by J. Kurzweil. The books contain a discussion of a gauge integral which is equivalent to the Perron integral. See also the books ”Theory of Integration” (1963; Zbl 0154.050) and ”Linear Analysis” (1967; Zbl 0172.390) by R. Henstock.}

Prof. McShane asks the following question: ”Can we take advantage of the close resemblance to Riemann integration to produce a unified theory that can be taught to students who have no intention of becoming research mathematicians, with just about the same level of difficulty as is encountered in ordinary courses, and that can also go from beginning calculus to the graduate level without ever abandoning earlier work and starting again (as usually now happens when Lebesgue integration is met)?” This text is an attempt to answer this question in the affirmative, and, in this reviewer’s opinion, it is a resounding success.

In Chapters 1 and 2, the gauge integral in one-dimension is introduced and its basic properties, including the Monotone and Dominated Convergence Theorems, are dicussed in detail. Chapter 4 discusses gauge integration in n-dimensions and includes versions of the Fubini and Tonelli theorems. Prof. McShane’s presentation of the gauge integral is painstakingly thorough and clear; he assumes minimal background on the part of the reader and presents proofs in complete detail. The presentation is clearly aimed at a reader with minimal background and preparation.

There is a very extensive list of applications presented in the text. Chapter 3 contains applications of the gauge integral to ordinary differential equations (its birth place) and probability. Chapter 5 contains discussions of both line and surface integrals. Chapter 6 contains applications to various topics in harmonic analysis; for example, \(L^ 1\) and \(L^ 2\), Hilbert spaces, orthogonal expansions and Fourier transforms. All of these discussions are very complete and thorough. Finally, Chapter 7 contains a dicussion of the Lebesgue integral with respect to an abstract measure. It is shown that this traditional approach to the Lebesgue integral is equivalent to the gauge integral in the case of Euclidean spaces. A number of additional topics, such as the Radon-Nikodym Theorem and Brownian motion, are also discussed.

Prof. McShane indicates his desire that the gauge integral will eventually become widely accepted and find its way into introductory analysis textbooks. While this book probably contains far too much material on integration and does not contain many other topics which are normally discussed in an introductory analysis course, it certainly does contain material from which a presentation of the gauge integral at an introductory level could be based. Hopefully, the mathematical community will accept Prof. McShane’s challenge and begin to incorporate a presentation of the gauge integral into introductory courses in real analysis.

{Reviewer’s remark: Several books have recently appeared which contain discussions of gauge-type integrals: ”The Generalized Riemann Integral” (1980; Zbl 0486.26005) by R. M. McLeod, ”Introduction à l’analyse” (1979; Zbl 0444.26002) by J. Mawhin, and ”Nichtabsolut konvergente Integrale” (1980; Zbl 0441.28001) by J. Kurzweil. The books contain a discussion of a gauge integral which is equivalent to the Perron integral. See also the books ”Theory of Integration” (1963; Zbl 0154.050) and ”Linear Analysis” (1967; Zbl 0172.390) by R. Henstock.}

Reviewer: Ch.Swartz

### MSC:

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

28A25 | Integration with respect to measures and other set functions |

26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |

28A15 | Abstract differentiation theory, differentiation of set functions |

60J65 | Brownian motion |

28A35 | Measures and integrals in product spaces |