# On the T-integration of Kartak and Marik

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

#### Mathematical reviews subject classification: 26A39, 26A45.

## Abstract

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.