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On extensions of the Riemann and Lebesgue Integrals by Nets

Published online by Cambridge University Press:  20 November 2018

O. S. Bellamy  and
H. W. Ellis
O. S. Bellamy
Affiliation:
University of the West Indies Cave Hill, Barbados
H. W. Ellis
Affiliation:
Queen's University Kingston, Ontario, Canada
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In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 1 , April 1975 , pp. 7 - 17
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Bellamy, O. S. and Ellis, H. W., On extension of the Riemann and Lebesgue integrals by nets, Queen's Mathematical Preprints, No. 1972–28.Google Scholar
2. Kelly, J. L., General Topology, D. Van Nostrand, New York, 1955.Google Scholar