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A Characterization of the Approximately Continuous Denjoy Integral

Published online by Cambridge University Press:  20 November 2018

Yöto Kubota
Yöto Kubota*
Affiliation:
Ibaraki University, Mito, Japan
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The approximately continuous integral which includes the Lebesgue integral has been considered by Burkill [1] and Ridder [3; 4]. I [2] also defined the AD-integral of this kind which is more general than the AP-integral of Burkill [1] and the Denjoy integral in the wide sense. But this integral is equivalent to the β-integral of Ridder.

Our aim in this paper is to characterize the AD-integral in the following way: The AD-integral is the least general approximately continuous integral (Definition 1) which includes the Lebesgue integral and fulfils the Cauchy and Harnack conditions (Definition 2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 219 - 226
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Burkill, J. C., The approximately continuous Perron integral, Math. Z. 34 (1931), 270–278.Google Scholar
2. Kubota, Y., An integral of the Denjoy type. III; III, Proc. Japan Acad. 40 (1964), 713-717; 42 (1966), 737-742; 43 (1967), 441–444.Google Scholar
3. Ridder, J., Ueber die gegenseitigen Beziehungen vershiedener approximate stetiger Denjoy-Perr on Intégrale, Fund. Math. 22 (1934), 136–162.Google Scholar
4. Ridder, J., Ueber approximate stetige Denjoy-Intégrale, Fund. Math. 21 (1933), 1–10.Google Scholar
5. Saks, S., Theory of the integral (Z. Subwencji Funduszu Kultury Narodowej, Warsaw; G. E. Stechert, New York, 1937).Google Scholar