On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative

The notion of $L^r$-variational measure generated by a function $F\in L^r[a,b]$ is introduced and, in terms of absolute continuity of this measure, a descriptive characterization of the $H\!K_r$-integral recovering a function from its $L^r$-derivative is given. It is shown that the class of functions generating absolutely continuous $L^r$-variational measure coincides with the class of $ACG_{r}$-functions which was introduced earlier, and that both classes coincide with the class of the indefinite $H\!K_{r}$-integrals under the assumption of $L^r$-differentiability almost everywhere of the functions consisting these classes.