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 be a brick in , and, for each brick and each vertex x of I, let h(I,x) be a real (or complex) number. The integral is defined to be a real (or complex) number H such that, for a given , there is a strictly positive function , defined on the closure of E, satisfying whenever E is the union of the mutually disjoint bricks is one of the vertices of , and 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 Fubini and Tonelli-type theorems for integrals in 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.