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Article 49

On the T-integration of Kartak and Marik

Real Analysis Exchange 28, no. 2 (2003), 515-542.

Mathematical reviews subject classification: 26A39, 26A45.



General notions of integration have been introduced by

  • S. Saks (see Theory of the integral, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937, p. 254),
  • K. Kartak (see A generalization of the Caratheodory theory of the differential equations, Czech. Math. J. 17 (1967), 482),
  • Y. Kubota (see A characterization of the Denjoy integral, Math. Japan. 26 (1981), p. 389) and 
  • D. N. Sarkhel (see A wide constructive integral Math. Japonica 32 (1987), p. 299).


Kartak's T integration was further studied by Kartak and Marik in the paper On representations of some Perron integrable functions ( Czech. Math. J. 19 (1969), 745-749), and by Kubota in the paper Notes on integration (Math. Japan. 31 (4) (1986), 617-621).


In this paper, starting from Kartak and Marik's definition, we introduce another general integration (see Definition 3.3), that allows a very general theorem of dominated convergence (see Theorem 3.1).

Then we present a general definition for primitives, and this definition contains many of the known nonabsolutely convergent integrals:

  • the Denjoy*-integral,
  • the α-Ridder integral,
  • the wide Denjoy integral,
  • the β-Ridder integral,
  • the Foran integral,
  • the AF integral,
  • the Gordon integral.


Using this integration and Theorem 3.1 we obtain a generalization of a result on differential equations, of Bullen and Vyborny (see P. S. Bullen and R. Vyborny, Some applications of a theorem of Marcinkiewicz, Canad. Math. Bull. 34 (2) (1991), 165-174).


We further give a Banach-Steinhaus type theorem, a categoricity theorem, Riesz type theorems (as a particular case we obtain the Alexiewicz Theorem (see A. Alexiewicz, Linear functionals on Denjoy integrable functions, Coll. Math. 1 (1948), 289-293), and study the weak convergence for the T-integration.