On the Lr-differentiability of two Lusin classes and a full descriptive characterization of the HKr-integral

Abstract

It is proved that any function of a Lusin-type class, the class of ACGr-functions, is differentiable almost everywhere in the sense of a derivative defined in the space Lr, 1 ≤ r < ∞. This leads to a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKr-integral, which serves to recover functions from their Lr-derivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that continuous ACG-functions can fail to be Lr-differentiable almost everywhere.