# Hake-Alexandroff-Looman type theorems

### Real Analysis Exchange 23 (1997/8), no. 2, 491-524.

#### Mathematical reviews subject classification: 26A39, 26A45, 26A46.

## Abstract

To define descriptive type integrals on compact intervals the following two facts are essential

1) To have a sufficiently general monotonicity theorem;

2) To find some linear spaces, sufficiently general, such that the monotonicity theorem can be applied.

In this paper we shall study three kinds of descriptive type integrals, that all generalize the wide Denjoy integral. In fact the classes of primitives for these integrals, restricted to the continuous functions, are ACG.

The integrals mentioned above are based on the following facts:

(I) [ACG] on a compact set is a linear space;

(II) [VBG] ∩ (N) on a compact set is a linear space (see D.N. Sarkhel and B.Kar, *(PVB) functions and integration*, J. Austral. Math. Soc. (Series A) **36** (1984), 335-353; Article 26; Article 33).

(III) VBG ∩ (N) for Borel functions on a Borel set is a linear space (Article 26; Article 33).

In *An analogue of the theorem Hake-Alexandroff-Looman *(Fund. Math. **C** (1978), 69-74), C. M. Lee introduced the very abstract LDG-integral, using (I) and his monotonicity Theorem A, a).

The integrals based on (II) and (III) use Theorem A, b).

To define Perron type integrals on compact intervals the following two facts are essential:

1') To have a sufficiently general monotonicity theorem;

2') To find some upper semilinear spaces, sufficiently general, such that the monotonicity theorem can be applied.

In this paper we shall study five kinds of Perron type integrals, that are all in a close relationship with the descriptive type integrals. These integrals are based on the following facts:

(I') [ACG], [LG] and [(AC ∩ LG] on a compact set are upper semilinear spaces;

(II') [VBG] ∩ (N) on a compact set is an upper semilinear space (Article 33);

(III') VBG ∩ (N) for Borel functions on a Borel set is an upper semilinear space (Article 33).

In *An analogue of the theorem Hake-Alexandroff-Looman *(Fund. Math. **C** (1978), 69-74), C. M. Lee introduced the very abstract LPG-integral, using [ACG] and Theorem A, a).

The Perron type integrals based on (II') and (III') use Theorem A, c).

The Hake-Alexandroff-Looman Theorem asserts that the restricted Denjoy integral is equivalent to the classical Perron integral (see S. Saks, *Theory of the integral*, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937, pp. 247-252).

In what follows, by a Hake-Alexandroff-Looman type theorem we mean a theorem that establishes the equivalence between a descriptive type integral and a Perron type integral.

In the last three sections we show some relationships between the descriptive type integrals and the Perron type integrals.

We obtain in Corollary 6 that C. M. Lee's LPG integral is a strict generalization of his LDG integral (although he claimed in *An analogue of the theorem Hake-Alexandroff-Looman *(Fund. Math. **C** (1978), 69-74), that they were equivalent).

In general the descriptive integrals (II) are strictly contained in the Perron type integrals (II'), but we identify situations so that the two integrals are equivalent.

We show that the descriptive integrals (III) and the Perron type integrals (III') are always equivalent.

Surprisingly, some descriptive integrals (II) are contained in some Perron type integrals (I').

It seems that the LPG integral cannot be characterized nicely descriptively.

However we identify two situations (see Definition 15) for which the LPG integral admits descriptive characterizations.