Geodesic
A Characterization of the Approximately Continuous Denjoy Integral
Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 219-226

Voir la notice de l'article provenant de la source Cambridge University Press

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). 
Kubota, Yöto. A Characterization of the Approximately Continuous Denjoy Integral. Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 219-226. doi: 10.4153/CJM-1970-028-0
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[1] 1. Burkill, J. C., The approximately continuous Perron integral, Math. Z. 34 (1931), 270–278. Google Scholar

[2] 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] 3. Ridder, J., Ueber die gegenseitigen Beziehungen vershiedener approximate stetiger Denjoy-Perr on Intégrale, Fund. Math. 22 (1934), 136–162. Google Scholar

[4] 4. Ridder, J., Ueber approximate stetige Denjoy-Intégrale, Fund. Math. 21 (1933), 1–10. Google Scholar

[5] 5. Saks, S., Theory of the integral (Z. Subwencji Funduszu Kultury Narodowej, Warsaw; G. E. Stechert, New York, 1937). Google Scholar

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