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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 0090.30002)] 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.

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 |