Lanzhou lectures on Henstock integration. (English) Zbl 0699.26004

Lee, Peng-Yee

Series in Real Analysis, 2. London etc.: World Scientific. viii, 179 p. £31.00/hbk; £15.00/pbk (1989).

The purpose of this book is to give an introduction to the theory of one- dimensional non-absolute integration including quite recent results and a survey on different methods of setting up the theory. The first two chapters present the Kurzweil-Henstock integral and its equivalence to the Denjoy and Perron integrals (both being defined in a descriptive manner). Chapter 3 contains Riesz type theorems on the representation of linear or orthogonally additive functionals on the Denjoy space (i.e. the linear space of all Denjoy integrable functions on [a,b] with the norm \(\| f\| =\sup \{| \int^{x}_{a}f|:\) \(a\leq x\leq b\}\); the main difficulty is that this space is not complete. Chapter 4 presents other methods leading to various integrals related to the Kurzweil- Henstock one, while, in Chapter 5, the reader finds sketches of more general theories. By studying any kind of integral, a central role is given to variants of the controlled convergence theorem. An extensive bibliography and some historical remarks are added.

Reviewer: Á.Császár

MSC:

26A39	Denjoy and Perron integrals, other special integrals
26-02	Research exposition (monographs, survey articles) pertaining to real functions
28-02	Research exposition (monographs, survey articles) pertaining to measure and integration

Keywords:

Kanugi integral; Perron integral; Denjoy integral; Cesàro-Perron integral; variational integral; McShane integral; introduction to the theory of one-dimensional non-absolute; integration; Kurzweil-Henstock integral; controlled convergence theorem; introduction to the theory of one-dimensional non-absolute integration