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The proximally continuous integrals

Part of: Real functions

Published online by Cambridge University Press:  09 April 2009

D. N. Sarkhel  and
A. K. De
D. N. Sarkhel
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India
A. K. De
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India
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Abstract

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This paper introduces certain generalization of the notions of approximate limit, continuity and derivative and of absolute continuity, of real functions, leading to generalized integrals of Perron and Denjoy types comprising the AP-integral of Burkill (1931) and Sonouchi and Utagawa (1949) and the AD-integral of Kubota (1963), respectiely. The generalizations are all substantiated by appropriate examples.

Keywords

AP-integralAD-integralapproximately continuousdensitysparse setproximally continuousDarboux continuityproximally absolutely continuousLusin's condition (N)PPr-integralPPl-integralPP-integralPD-integral

MSC classification

Secondary: 26A03: Foundations 26A24: Differentiation (functions of one variable) 26A39: Denjoy and Perron integrals, other special integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 1 , June 1981 , pp. 26 - 45
Copyright
Copyright © Australian Mathematical Society 1981

References

Bauer, H. (1915), ‘Der Perronsche Integralbegriff und seine Beziehung zum Lebesgueschen’, Monatsh. Math. Phys. 26, 153–198.Google Scholar
Burkill, J. C. (1931), ‘The approximately continuous Perron Integral’, Math. Z. 34, 270–278.Google Scholar
Denjoy, A. (1916), ‘Sur la derivation et son calcul inverse’, C. R. Acad. Sci. Paris Sér. A-B 162, 377–380.Google Scholar
Hake, H. (1921), ‘Über de la Vallée Poussins Ober-und Unterfunktionen einfacher Integrale und die Integral-definition von Perron’, Math. Ann. 83, 119–142.Google Scholar
Kennedy, M. D. (1930–1931), ‘Upper and lower Lebesgue Integrals’, Proc. London Math. Soc. (2) 32, 21–50.Google Scholar
Kubota, Yoto (1963), ‘On the approximately continuous Denjoy Integrals’, Tôhoku Math. J. 15, 253–264.Google Scholar
Saks, S. (1937), Theory of the integral (second revised edition, Warsaw, 1937; reprint, Dover Publ., New York, 1964).Google Scholar
Sonouchi, G. and Utagawa, M. (1949), ‘The generalized Perron Integrals’, Tôhoku Math. J. 1, 95–99.Google Scholar