# On an improvement of the Hake theorem

### Real Analysis Exchange 24 (1998/9), No. 2, 867-872.

#### Mathematical reviews subject classification: 26A39, 26A42, 26A46.

## Abstract

The well-known Hake Theorem asserts that:

If a function f is Denjoy* integrable then it is also Perron integrable, and the two integrals are equal.

In fact these two integrals are equivalent (see the Hake-Alexandroff-Looman Theorem), and there are many definitions of Perron-type integrals that are equivalent to the Denjoy* integral.

In Corollary 5.9.1 of Real Functions - Current Topics we made a study of many (at least 108) of these equivalences.

One of the strongest Perron type definition is that of Skljarenko, where the major and minor functions are AC*G and continuous.

Using the Tolstoff-Zahorski Theorem we showed that in addition, the major and minor functions have finite or infinite derivatives at each point, obtaining the-integral.

To prove that the Denjoy*-integrability implies the - integrability (i.e., a Hake type theorem) we used essentially the Vitali-Caratheodory Theorem (see I. P. Natanson, *Theory of functions of a real variable*, 2nd. rev. ed., Ungar, New York, 1961, p. 166).

In the present paper we give a different, less technical proof of this result, using essentially Lusin's Theorem (see S. Saks, *Theory of the integral*, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937, p. 72).

Both proofs use different techniques from that of Skljarenko.

We conclude the paper with some comments and a question related to the subject.