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A function is Perron integrable if it has locally small Riemann sums

Part of: Real functions

Published online by Cambridge University Press:  09 April 2009

Arlo W. Schurle
Arlo W. Schurle
Affiliation:
Department of Mathematical Sciences, University of Petroleum and Minerals, UPM Box 639, Dhahran 31261, Saudi Arabia
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Abstract

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We say that a function has locally small Riemann sums on an interval if for each point x in the interval, and for each positive number ε, all sufficiently fine partitions of intervals lying in neighborhood of x but not containing x have Riemann sums of absolute value less than ε. The main result is then as the title states. We use the generalized Riemann approach to Perron integration, assuming that functions are measurable only to insure that conditions involving the positive and negative parts of the functions are satisfied.

MSC classification

Secondary: 26A39: Denjoy and Perron integrals, other special integrals 26A42: Integrals of Riemann, Stieltjes and Lebesgue type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 2 , October 1986 , pp. 224 - 232
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]McLeod, R. M., The generalized Riemann integral (Carus Mathematical Monograph No. 20, American Mathematical Association, Washington, D. C., 1980).CrossRefGoogle Scholar
[2]Natanson, I. P., Theory of functions of a real variable, Vol. II (translated by Boron, L. F., Frederick Ungar Publishing Co., New York, 1960).Google Scholar
[3]Pfeffer, W. F., ‘The Riemann-Stieltjes approach to integration’ (Technical Report 187, National Research Institute for Mathematical Sciences, Pretoria, South Africa, 1980).Google Scholar
[4]Saks, S., Theory of the integral (2nd revised edition, translated by Young, L. C., Dover Publications, New York, 1964).Google Scholar
[5]Schurle, A., ‘Perron integrability versus Lebesgue integrability’, Canad. Math. Bull. 28 (1985), 463–468.CrossRefGoogle Scholar