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The Burkill approximately continuous integral

Part of: Real functions

Published online by Cambridge University Press:  09 April 2009

P. S. Bullen
P. S. Bullen
Affiliation:
Mathematics DepartmentThe University of British ColumbiaVancouver, British Columbia, Canada
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Abstract

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This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill.

MSC classification

Secondary: 26A39: Denjoy and Perron integrals, other special integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 2 , October 1983 , pp. 236 - 253
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Aleksandrov, A., ‘Über die Äquivalenz des Perronschen und des Denjoyschen Integralbegriffes’, Math. Z. 20 (1924), 213–222.Google Scholar
[2]Bosanquet, L. S., ‘A property of Cesáro-Perron integrals’, Proc. Edinburgh Math. Soc. (2) 6 (1940), 160–165.CrossRefGoogle Scholar
[3]Bullen, P. S., ‘Non-absolute integrals; a survey’, Real Anal. Exchange 5 (1980), 195–259.CrossRefGoogle Scholar
[4]Burkill, J. C., ‘The approximately continuous Perron integral’, Math. Z. 34 (1931), 270–278.CrossRefGoogle Scholar
[5]Grimshaw, M. E., ‘The Cauchy property of the generalised Perron integrals’, Proc. Cambridge Philos. Soc. 30 (1934), 15–18.CrossRefGoogle Scholar
[6]Henstock, R., Theory of integration (London, 1963).Google Scholar
[7]Henstock, R., Linear analysis (London, 1967).Google Scholar
[8]Henstock, R., ‘The variation on the real line’, Proc. Roy. Irish Acad. Sect. A 79 (1979), 1–10.Google Scholar
[9]Kubota, Y., ‘An integral of Denjoy’, I. Proc. Japan. Acad. 40 (1964), 713–717.Google Scholar
[10]Kubota, Y., ‘On the compatibility of the AP-and the D-integrals’, Proc. Japan Acad. 44 (1968), 330–333.Google Scholar
[11]Kubota, Y., ‘A characterization of the approximately continuous Denjoy integral’, Canad. J. Math. 22 (1970), 219–226.CrossRefGoogle Scholar
[12]Kubota, Y., ‘A constructive definition of the approximately continuous Denjoy integral’, Canad. Math. Bull. 15 (1972), 103–108.CrossRefGoogle Scholar
[13]Kubota, Y., ‘A direct proof that the RC-integral is equivalent to the D*-integral’, Proc. Amer. Math. Soc. 80 (1980), 293–296.Google Scholar
[14]Kubota, Y., ‘An elementary theory of the special Denjoy integral’, Math. Japon. 24 (1980), 507–520.Google Scholar
[15]O'Malley, R., ‘Selective derivatives’, Acta Math.Acad. Sci. Hungar. 29 (1977), 77–79.CrossRefGoogle Scholar
[16]Pfeffer, W. F., The Riemann-Stieltjes approach to approach to integration (Technical Report, TWISK 187, National Research Inst. for Math. Sci., Pretoria, 1980).Google Scholar
[17]Ridder, J., ‘Über den Perronschen Integralbegriff und seine Beziehung zu den R-, L- und D-Integralen’, Math. Z. 34 (1932), 234–269.CrossRefGoogle Scholar
[18]Ridder, J., ‘Über approximativ stetige Denjoy-Integrale’, Fund. Math. 21 (1933), 1–10.CrossRefGoogle Scholar
[19]Ridder, J., ‘Über die Gegenseitigen Beziehungen verschiedener approximativ stetiger Denjoy-Perron Integrale’, Fund. Math. 22 (1934), 136–162.CrossRefGoogle Scholar
[20]Saks, S., Theory of the integral, 2nd ed., revised (New York, 1937).Google Scholar
[21]Solomon, D. W., Denjoy integration in abstract spaces (Mem. Amer. Math. Soc. 85, 1969).Google Scholar
[22]Sunouchi, G. and Utagawa, M., ‘The generalized Perron integrals’, Tôhoku Math. J. 1 (1949), 95–99.CrossRefGoogle Scholar
[23]Thomson, B. S., ‘On full coverings’, Real Anal. Exchange 6 (1980–1981), 77–93.CrossRefGoogle Scholar
[24]Tolstov, G. P., ‘Sur quelques propriétés des fonctions approximativement continues’, Mat. Sb. 5 (1939), 637–645.Google Scholar
[25]Tolstov, G. P., ‘Sur l'intégrale de Perron’, Mat. Sb. 5 (1939), 647–659.Google Scholar
[26]Tolsov, G. P., ‘La méthode de Perron pour l'intégrale de Denjoy’, Mat. Sb. 8 (1940), 149–167.Google Scholar
[27]Zahorski, Z., ‘Ueber die Menge der Punkte in welchen die Ableitung unendlich ist’, Tôhoku Math. J. 48 (1941), 321–332.Google Scholar