On the $L^r$-differentiability of Two Lusin Classes and a Full Descriptive Characterization of the $H{K}_r$-integral

It is proved that any function of a Lusin-type class, the class of $AC{G}_r$-functions, is differentiable almost everywhere in the sense of a derivative defined in the space~$L^r$, $1\le r<\mathrm{\infty }$. This leads to obtaining a full descriptive characterization of a Henstock-Kurzweil-type integral, the $H{K}_r$-integral, which serves to recover functions from their $L^r$-derivatives. The class $AC{G}_r$ is compared with the classical Lusin class $ACG$ and it is shown that a continuous $ACG$-function can fail to be $L^r$-differentiable almost everywhere.