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Zbl 1024.26004
Bongiorno, Benedetto
The Henstock-Kurzweil integral. (English)
[A] Pap, E. (ed.), Handbook of measure theory. Vol. I and II. Amsterdam: North-Holland. 587-615 (2002). ISBN 0-444-50263-7/hbk

The paper is an extremely useful survey of the Henstock-Kurzweil integration theory which started in 1957 by the work of Jaroslav Kurzweil and independently in 1961 by the work of Ralph Henstock. The original theory was build up on an approach to integration based on Riemann sums while the resulting integral is equivalent to the narrow Denjoy integral, i.e., it allows to integrate every derivative and it includes the Lebesgue integral. The main idea is based on a modified approach to the concept of fineness of partitions of an interval. In this work definitions and basic properties of the Henstock-Kurzweil integral are presented, attention is paid to various modifications of the integration theory over one-dimensional intervals and to integrals in the case of multidimensional intervals over which functions are integrated. The multidimensional integration domain is interesting because using the original one-dimensional definition we obtain an integral which does not integrate the divergence of any differentiable vector field and also the usual transformation formula fails in general. Therefore the dedfinition of the Riemann-type integral was modified in many directions in order to keep the nice properties also for the multidimensional case. An account of these efforts is presented in the survey. Integration of vector valued functions and integration over general spaces are shortly mentioned at the end of the paper. A very good job was done by the author in this survey. The main contemporary results are described in a clear way (without proofs, of course) and a very good and fairly complete bibliography is appended to the paper where the detailed information can be found.
[Stefan Schwabik (Praha)]

MSC 2000:: *26A39 Special integrals of functions of one real variable
26B15 Integration (several real variables)
28A25 Integration with respect to measures and other set functions

Keywords: Henstock-Kurzweil integral; survey; Denjoy integral; Lebesgue integral; Riemann-type integral; multidimensional integration

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