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 \(E=\prod^{n}_{1}[a_ i,b_ i)\) 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 \(\epsilon >0\), there is a strictly positive function \(\delta\), defined on the closure of E, satisfying \(| s-H| <\epsilon\) whenever \(s=\sum^{n}_{1}h(I_ k,x_ k),\) E is the union of the mutually disjoint bricks \(I_ 1,...,I_ m,\) \(x_ k\) is one of the vertices of \(I_ k\), and \(diam I_ k<\delta (x_ k).\) 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 \(H(E)=\int_{E}f d\mu,\) 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.

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 \(\epsilon >0\), there is a strictly positive function \(\delta\), defined on the closure of E, satisfying \(| s-H| <\epsilon\) whenever \(s=\sum^{n}_{1}h(I_ k,x_ k),\) E is the union of the mutually disjoint bricks \(I_ 1,...,I_ m,\) \(x_ k\) is one of the vertices of \(I_ k\), and \(diam I_ k<\delta (x_ k).\) 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 \(H(E)=\int_{E}f d\mu,\) 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: Á.Császár

##### 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 |