*(English)*Zbl 0668.28001

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 $E={\prod}_{1}^{n}[{a}_{i},{b}_{i})$ 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 $\u03f5>0$, there is a strictly positive function $\delta $, defined on the closure of E, satisfying $|s-H|<\u03f5$ whenever $s={\sum}_{1}^{n}h({I}_{k},{x}_{k}),$ E is the union of the mutually disjoint bricks ${I}_{1},\xb7\xb7\xb7,{I}_{m},$ ${x}_{k}$ is one of the vertices of ${I}_{k}$, and $diam{I}_{k}<\delta \left({x}_{k}\right)\xb7$ 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.

##### 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 |