Geodesic
On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative
Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 411-421

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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.
Keywords: $L^r$-derivative, Henstock–Kurzweil-type integral, $L^r$-variational measure, absolutely continuous measure, generalized absolute continuity of a function. 
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     title = {On {Descriptive} {Characterizations} of an {Integral} {Recovering} a {Function} from {Its} $L^r${-Derivative}},
     journal = {Matemati\v{c}eskie zametki},
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P. Musial; V. A. Skvortsov; F. Tulone. On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative. Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 411-421. http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a7/