Lectures on the theory of integration. (English) Zbl 0668.28001

Henstock, Ralph

Series in Real Analysis, 1. Singapore: World Scientific Publishing Co.. xii, 206 p. £53.00 (1988).

The purpose of this book is to give a detailed study of the theory of integration together with some applications, based on the definition due to J. Kurzweil [Czech. Math. J. 7(82), 418–446 (1957; Zbl 0090.30002)] and to the author [J. Lond. Math. Soc. 30, 273–286 (1955; Zbl 0066.09204); Proc. Lond. Math. Soc. (3) 11, 402–418 (1961; Zbl 0099.27402)].
In spite of its generality, the definition is surprisingly simple and elementary. Let \(E=\prod^n_1[a_i,b_i)\) be a brick in \(R^n\), and, for each brick \(I\subset E\) and each vertex \(x\) of \(I\), let \(h(I,x)\) be a real (or complex) number. The integral \(\int_{E}\,dh\) is defined to be a real (or complex) number \(H\) such that, for a given \(\varepsilon >0\), there is a strictly positive function \(\delta\), defined on the closure of \(E\), satisfying \(| s-H| <\varepsilon\) whenever \(s=\sum^n_1 h(I_k,x_k)\), \(E\) is the union of the mutually disjoint bricks \(I_1,\ldots,I_m\), \(x_k\) is one of the vertices of \(I_k\), and \(\operatorname{diam} I_k<\delta (x_k)\). It is shown that this integral embraces, among others, those of Riemann-Stieltjes, Burkill, Lebesgue, Denjoy-Perron, and that all essential properties of the Lebesgue integral can be, under suitable conditions and a suitable form, generalized to it. So integration of sequences of functions, derivation with respect to a parameter, differentiation of the integral function \(H(E)=\int_{E}f \,d\mu\), Fubini and Tonelli-type theorems for integrals in \(R^{m+n}\) are discussed.
Applications in the theory of ordinary differential equations, probability theory and statistics are added.
The book ends with a careful survey on relations with other integrals and with detailed historical remarks.

Reviewer: Ákos Császár (Budapest)

MSC:

28-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
28A25	Integration with respect to measures and other set functions
26A39	Denjoy and Perron integrals, other special integrals
26A42	Integrals of Riemann, Stieltjes and Lebesgue type
28A10	Real- or complex-valued set functions
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
28A35	Measures and integrals in product spaces

Keywords:

\(\delta\)-fine division; generalized Riemann integral; brick-point function; variation set; Riemann-Stieltjes integral; Burkill integral; Denjoy-Perron integral; Lebesgue integral; Fubini and Tonelli-type theorems; ordinary differential equations; probability theory; statistics

Citations:

Zbl 0090.30002; Zbl 0066.09204; Zbl 0099.27402