Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-07T16:43:45.333Z Has data issue: false hasContentIssue false
  • English
  • Français

A Riemann integral in a locally compact Hausdorff space

Part of: Set functions and measures on spaces with additional structure Real functions Classical measure theory Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  09 April 2009

S. I. Ahmed  and
W. F. Pfeffer
S. I. Ahmed
Affiliation:
Department of Mathematics, King Saud UniversityRiyadh 11451, Saudi Arabia
W. F. Pfeffer
Affiliation:
Department of Mathematics, University of California, Davis, California 95616, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

MSC classification

Secondary: 28A25: Integration with respect to measures and other set functions 26A42: Integrals of Riemann, Stieltjes and Lebesgue type 54C30: Real-valued functions 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 1 , August 1986 , pp. 115 - 137
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Henstock, R., Theory of integration (Butterworths, London, 1963).Google Scholar
[2]McLeod, R. M., The generalized Riemann integral (MAA, Providence, 1980).CrossRefGoogle Scholar
[3]McShane, E. J., ‘A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals’, Mem. A mer. Math. Soc. 88 (1969).Google Scholar
[4]McShane, E. J., ‘A unified theory of integration’, Amer. Math. Monthly 80 (1973), 349–359.CrossRefGoogle Scholar
[5]Pfeffer, W. F., ‘On the lower derivative of a set function’, Canad. J. Math. 20 (1968), 1489–1498.CrossRefGoogle Scholar
[6]Pfeffer, W. F., ‘An integral in topological spaces I’, J. Math. Mech. 18 (1969), 953–972.Google Scholar
[7]Pfeffer, W. F., ‘An integral in topological spaces II’, Math. Scand. 27 (1970), 77–104.CrossRefGoogle Scholar
[8]Pfeffer, W. F., Integrals and measures (Dekker, New York, 1977).Google Scholar
[9]Pfeffer, W. F., ‘The Riemann-Stieltjes approach to integration’, TWISK 187 (CSIR: NRIMS. Pretoria, 1980).Google Scholar