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Generalized Riemann Integration and an Intrinsic Topology

Published online by Cambridge University Press:  20 November 2018

Ralph Henstock
Ralph Henstock*
Affiliation:
The New University of Ulster, Northern Ireland
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In generalized Riemann integration theory it is becoming increasingly clear that a particular collection of sets has some properties of a topology; it is a useful topology when general requirements hold, and the present paper examines the background. Thomson [23, 24] altered my original theory of the variation and Riemann-type integration that has Lebesgue properties, defining the variation of a function of interval-point pairs over the whole of a space T by using partial divisions of T instead of divisions covering T entirely, and also defining a Lebesgue-type integral. His reason might have been that a decomposable division space seems impossible in a general compact or locally compact space. McGill mentioned this to me, and in [15] connected Thomson's setting with topological measure and Topsøfe [25], giving an interesting theorem on the variation of the limit of a monotone increasing generalized sequence of open sets.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 395 - 413
Copyright
Copyright © Canadian Mathematical Society 1980

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