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Article 5

Nonabsolute convergent integrals

Real Analysis Exchange 11 (1986), 121-134

Mathematical reviews subject classification: 26A39



Interesting generalizations of the descriptive definition for nonabsolutely convergent integrals were given by H. W. Ellis (see Darboux Properties and Applications to non-absolute convergent  integrals, Can. J. Math., 3 (1951), 471-485), J. Foran (see An extension of the Denjoy integral, Proc. Amer. Math. Soc., 49 (1975), 359-365) and C. M. Lee (see An analogue of the theorem Hake-Alexandroff-Looman, Fund. Math. C (1978), 69-74).

The most remarkable one is that of Foran, which is a classical generalization of the Denjoy integral in the wide sense, i.e., Foran's class of primitives is a class of continuous functions which contains strictly the ACG functions.  

The classes of primitives for the integrals of Ellis and the integrals of Lee are not classes of continuous functions, and restricted to the class of continuous functions one obtains at most the class of ACG functions.  

In this paper we give various extensions for each of these integrals. The classes of primitives for our generalizations are not classes of continuous functions. However, if one restricts these primitives to the continuous functions, some of these classes contain strictly the primitives in the Foran sense.

The uniqueness of the integration for the Foran integral follows by a corollary of Theorem 7.7 of S. Saks (Theory of the integral, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937).

 To assure the uniqueness of our integrations, we give some monotonicity theorems among which Theorem 3 is the most important.

Theorem 3 generalizes Theorem 7.7 of Saks, and its corollaries are both intrinsically interesting and useful.