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A modern theory of integration. (English) Zbl 0968.26001
Graduate Studies in Mathematics 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0845-1). xiv, 458 p. (2001).
This monograph is a comprehensive, beautifully written exposition of the Henstock-Kurzweil (gauge, Riemann complete) integral. This integral, which was discovered independently in the late 1950’s by R. Henstock and J. Kurzweil, is obtained by making a “slight” alteration in the usual definition of the Riemann integral. Let \(f:[a,b]\rightarrow \mathbb{R}\) ; the function \(f\) is Riemann integrable if there exists \(A\) such that for every \(\varepsilon >0\) there exists a \(\delta >0\) such that whenever \( \{a=x_{0}<x_{1}<\dots <x_{n}=b\}\) is a partition of \([a,b]\) with \(\max (x_{i+1}-x_{i})<\delta \) and \(x_{i}\leq t_{i}\leq x_{i+1}\), then \(|\sum_{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-A|<\varepsilon \). In the Henstock-Kurzweil (H-K) integral the constant \(\delta \) in Riemann’s definition is essentially replaced by a positive function \(\delta \). The change in the resulting integral is indeed profound. The H-K integral is more general than the Lebesgue integral, in fact is equivalent to the integrals of Perron and Denjoy, the Fundamental Theorem of Calculus holds in full generality for the integral, the Monotone and Dominated Convergence Theorems are valid and there are no “improper” integrals so the integral admits conditionally convergent integrals.
Part 1 of the book contains a very careful and detailed exposition of the H-K integral over a compact interval in the real line. Using only elementary methods the author develops the basic properties of the integral including the Monotone and Dominated Convergence Theorems and the Fundamental Theorem of Calculus in full generality. The author uses the H-K integral to define and study the properties of Lebesgue measure and Lebesgue measurable functions. Absolute integrability, absolute continuity and mean convergence, including the Riesz-Fischer Theorem, are treated in detail.
Part 2 of the book extends the H-K integral to unbounded intervals. Previous results which can be easily extended to the case of unbounded intervals are clearly indicated while detailed proofs of results which require significant changes are supplied.
There is an abundant supply of exercises which serve to make this book an excellent choice for a text for a course which would contain an elementary introduction to modern integration theory.

26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
26A39 Denjoy and Perron integrals, other special integrals
28A25 Integration with respect to measures and other set functions