##
**The general theory of integration.**
*(English)*
Zbl 0745.26006

Oxford Mathematical Monographs. Oxford: Clarendon Press. xi, 262 p. (1991).

About 40 years ago the theory of integration received a simple new idea, that, as in common with many such, was introduced independently, and almost simultaneously by more than one person. In this case the idea was introduced by Henstock in the United Kingdom and by Kurzweil in Czechoslovakia. It has been suggested that if Lebesgue had really been as involved in the search for primitives as the title of his famous book suggests, then he might well have discovered the same idea 50 years earlier, having the unfortunate result of delaying the development of measure theory. In fact he did not and mathematics has benefitted from his oversight. The simple idea is to allow the intervals in the Riemann sum to have sizes that depend on where they are in the interval of integration. In this way an integral is defined that is more general than the Lebesgue integral, to which it reduces when restricted to the class of absolutely integrable functions. Once this step has been made it is very natural to consider Riemann sums of the form \(\sum h(y_ i,[a_{i- 1},a_ i])\), where \(h\) is a real-valued point-interval function, and so include a wide class of Stieltjes and Burkhill integrals in the same theory. It has been found in the past that integrals that are more general than the Lebesgue integral fail to fit nicely into a general theory, it is even difficult to extend them to two dimensions. General theories exist, but on the whole give no insights, unlike general measure theory.

The present work is the basic one on an abstract theory of the general Riemann integral. It extends, revises and simplifies material found in two earlier books by the same author [“Theory of integration. London: Butterworth & Co.(1963; Zbl 0154.05001), now out of print, and “Lectures on the theory of integration.” (1988; Zbl 0668.28001)]. None of these books is easy to read but a knowledge of either, or both of the earlier books is essential for a full understanding of the present one, since they are less abstract. Even reading of the more standard treatments as in Lee Peng-Yee’s “Lanzhou lectures in Henstock integration.” London etc.: World Scientific (1989; Zbl 0699.26004), or R. M. McLeod’s “The generalized Riemann integral.” Washington, D. C.: The Mathematical Association of America (1980; Zbl 0486.26005), or E. J. McShane’s “Unified integration.” Orlando etc.: Academic Press (1983; Zbl 0551.28001) is advisable. A good feature of the present book is its numerous references to the literature, to papers where extra motivation is often provided for what are often very difficult ideas.

Particularly useful in this regard are the books and papers by B. S. Thomson, especially “Real functions.” [Lecture Notes in Mathematics. 1170. Berlin etc.: Springer-Verlag (1985; Zbl 0581.26001)] in the Lecture Notes series of Springer, the A.M.S. Memoir “Derivates of interval functions” [Mem. Am. Math. Soc. 45, 96 p. (1991; Zbl 0734.26003)], and two long papers in the Real Anal. Exch. 8, 62–207 (1983; Zbl 0525.26002), ibid. 8, 278–442 (1983; Zbl 0525.26003) which, although concerned with different topics, use many of the abstract ideas originating in Henstock’s work.

Henstock’s book develops ideas of great ingenuity and insight in an attempt to fit non-absolute integrals into a theory that, unlike most earlier theories includes many more integrals than the classical one of Denjoy and Perron. How useful this theory will be remains to be seen, but already many of the ideas in it have found uses, as for instance in the work of Thomson. A particular difficulty in fitting Riemann integrals into a general theory is the special role played by intervals, used to partition the domain of integration. Henstock’s abstract version of this is his idea of division spaces; another basic idea is that of a variational integral.

The definitions, too involved to give here, are developed in the first two chapters of the book, and give a general theory of integration with strong limit theorems, Chapter 3. The next chapter deals with differentiation – an area that has, as the author states, “yet to be clarified”. Chapter 5, 6 deal with product spaces, it is in the area of infinite product spaces that the author is presently devoting his energies, trying to fit the Feynman integral into his general approach to integration. Next the author considers Perron definitions, Ward definitions and convergence factor integrals (such as Cesàro-Perron integrals) in the setting of his theory. A final short chapter gives some functional analytic properties of the spaces of certain of his integrals (they form barrelled spaces) and a mention of density integration. All in all it is a valiant attempt to bring a difficult subject under control and should be known to all interested in integration.

The present work is the basic one on an abstract theory of the general Riemann integral. It extends, revises and simplifies material found in two earlier books by the same author [“Theory of integration. London: Butterworth & Co.(1963; Zbl 0154.05001), now out of print, and “Lectures on the theory of integration.” (1988; Zbl 0668.28001)]. None of these books is easy to read but a knowledge of either, or both of the earlier books is essential for a full understanding of the present one, since they are less abstract. Even reading of the more standard treatments as in Lee Peng-Yee’s “Lanzhou lectures in Henstock integration.” London etc.: World Scientific (1989; Zbl 0699.26004), or R. M. McLeod’s “The generalized Riemann integral.” Washington, D. C.: The Mathematical Association of America (1980; Zbl 0486.26005), or E. J. McShane’s “Unified integration.” Orlando etc.: Academic Press (1983; Zbl 0551.28001) is advisable. A good feature of the present book is its numerous references to the literature, to papers where extra motivation is often provided for what are often very difficult ideas.

Particularly useful in this regard are the books and papers by B. S. Thomson, especially “Real functions.” [Lecture Notes in Mathematics. 1170. Berlin etc.: Springer-Verlag (1985; Zbl 0581.26001)] in the Lecture Notes series of Springer, the A.M.S. Memoir “Derivates of interval functions” [Mem. Am. Math. Soc. 45, 96 p. (1991; Zbl 0734.26003)], and two long papers in the Real Anal. Exch. 8, 62–207 (1983; Zbl 0525.26002), ibid. 8, 278–442 (1983; Zbl 0525.26003) which, although concerned with different topics, use many of the abstract ideas originating in Henstock’s work.

Henstock’s book develops ideas of great ingenuity and insight in an attempt to fit non-absolute integrals into a theory that, unlike most earlier theories includes many more integrals than the classical one of Denjoy and Perron. How useful this theory will be remains to be seen, but already many of the ideas in it have found uses, as for instance in the work of Thomson. A particular difficulty in fitting Riemann integrals into a general theory is the special role played by intervals, used to partition the domain of integration. Henstock’s abstract version of this is his idea of division spaces; another basic idea is that of a variational integral.

The definitions, too involved to give here, are developed in the first two chapters of the book, and give a general theory of integration with strong limit theorems, Chapter 3. The next chapter deals with differentiation – an area that has, as the author states, “yet to be clarified”. Chapter 5, 6 deal with product spaces, it is in the area of infinite product spaces that the author is presently devoting his energies, trying to fit the Feynman integral into his general approach to integration. Next the author considers Perron definitions, Ward definitions and convergence factor integrals (such as Cesàro-Perron integrals) in the setting of his theory. A final short chapter gives some functional analytic properties of the spaces of certain of his integrals (they form barrelled spaces) and a mention of density integration. All in all it is a valiant attempt to bring a difficult subject under control and should be known to all interested in integration.

Reviewer: Peter S. Bullen (Vancouver)

### MSC:

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

26A39 | Denjoy and Perron integrals, other special integrals |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |