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Lectures on the theory of integration. (English) Zbl 0668.28001
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 0090.300)] and to the author [J. Lond. Math. Soc. 30, 273-286 (1955; Zbl 0066.092); Proc. Lond. Math. Soc., III. Ser. 11, 402-418 (1961; Zbl 0099.274)].

In spite of its generality, the definition is surprisingly simple and elementary. Let $E={\prod }_{1}^{n}\left[{a}_{i},{b}_{i}\right)$ 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 $ϵ>0$, there is a strictly positive function $\delta$, defined on the closure of E, satisfying $|s-H|<ϵ$ whenever $s={\sum }_{1}^{n}h\left({I}_{k},{x}_{k}\right),$ E is the union of the mutually disjoint bricks ${I}_{1},···,{I}_{m},$ ${x}_{k}$ is one of the vertices of ${I}_{k}$, and $diam{I}_{k}<\delta \left({x}_{k}\right)·$ It is shown that this integral embrasses, 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 $H\left(E\right)={\int }_{E}fd\mu ,$ 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: Á.Császár

##### MSC:
 28-01 Textbooks (measure and integration) 28A25 Integration with respect to measures and other set functions 26A39 Special integrals of functions of one real variable 26A42 Integrals of Riemann, Stieltjes and Lebesgue type (one real variable) 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