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A Characterization of the Approximately Continuous Denjoy Integral

  • Yöto Kubota (a1)

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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).

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References

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1. Burkill, J. C., The approximately continuous Perron integral, Math. Z. 34 (1931), 270278.
2. Kubota, Y., An integral of the Denjoy type. III; III, Proc. Japan Acad. 40 (1964), 713-717; 42 (1966), 737-742; 43 (1967), 441444.
3. Ridder, J., Ueber die gegenseitigen Beziehungen vershiedener approximate stetiger Denjoy-Perr on Intégrale, Fund. Math. 22 (1934), 136162.
4. Ridder, J., Ueber approximate stetige Denjoy-Intégrale, Fund. Math. 21 (1933), 110.
5. Saks, S., Theory of the integral (Z. Subwencji Funduszu Kultury Narodowej, Warsaw; G. E. Stechert, New York, 1937).
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A Characterization of the Approximately Continuous Denjoy Integral

  • Yöto Kubota (a1)

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