On an Improvement of the Hake Theorem

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 \cite{Ene1} we introduced a very strong Perron integration $({\mathcal P}_{1,1})$ and proved the corresponding Hake-type theorem, using the Vitali-Carath\'eodory Theorem. In this paper we give a new, less technical proof of this result, using essentially Lusin's Theorem.