A definition of integrals
type: Item
instance of: scholarly article
title: A definition of integrals
zbMATH Open document ID: 0846.26007
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-09-23T00:00:00Z
review text: In this paper a very general approach to Riemann integration is developed. The general idea is to associate with a real valued function Riemann sums depending on partitions that avoid small sets that contain the pathologies of the function. In order to do this the author needs families of partitions that are equipped with a filter, and a system of small sets that is a filter on the set of subsets of the space on which the functions are defined. The resulting definition of an integration space, a 6-tuple, and the associated integral are fairly abstract. Restrictions on the general theory give an absolute integral that is the Lebesgue integral with respect to a natural measure associated with the integration space. Another section considers the system of small sets defined by a gauge; and the resulting integral is the same as the associated gauge integral. In particular, an integral is obtained that is at least as general as the Kurzweil-Henstock integral. The concepts and definitions are too many and detailed to be discussed here but all those working in the theory of general integrals should read this very interesting paper. The full connections with the Henstock theory remain to be developed.
Mathematics Subject Classification ID: 26A39
Mathematics Subject Classification ID: 28A25
zbMATH DE Number: 861800
zbMATH Keywords: Riemann integration
zbMATH Keywords: Lebesgue integral
zbMATH Keywords: gauge integral
zbMATH Keywords: Kurzweil-Henstock integral
author: Nicolas Spaltenstein
reviewed by: P. S. Bullen
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1392
OpenAlex ID: W2074876829
DOI: 10.1006/JMAA.1995.1392
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description: scientific article; zbMATH DE number 861800