Abstract. We present a method of integration along the lines of the HenstockKurzweil integral. All L r -derivatives are integrable in this method.

We compare a Perron-type integral with a Henstock-Kurzweil-type integral, both having been introduced to recover functions from their generalized derivatives defined in the metric L r L^r . We give an example of an H K r HK_r -integrable function which is not P r P_r -integrable, thereby showing that the first integral is strictly wider than the second one.

We define the $$L^r$$ L r -variational integral and we prove that it is equivalent to the $$HK_r$$ H K r -integral defined in 2004 by P. Musial and Y. Sagher in the Studia Mathematica paper The$$L^{r}$$ L r -Henstock–Kurzweil integral. We prove also the continuity of $$L^r$$ L r -variation function.

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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 L r , 1 ≤ r < ∞. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the HKrintegral, which serves to recover functions from their L rderivatives. The class ACGr is compared with the classical Lusin class ACG and it is shown that a continuous ACGfunction can fail to be L r -differentiable almost everywhere.