On the Denjoy-Luzin definitions of the function classes ACG, ACG*, VBG, and VBG*

Absolute continuity with respect to a subset of an interval

“…In one-dimensional case, the articles are mainly devoted to definitions that are used for descriptive definition of Henstock-Kurzweil integral. There are several attempts to this notion; let us mention Luzin's and Denjoy's definitions of AC, ACG, AC * and ACG * or for continuous functions equivalent approach by Khintchine (see [7], [13] for definitions and [15] for the comparison of these approaches). In agreement with classical constructive definition of Henstock-Kurzweil integral, Gordon ([6]) introduced the notion of AC δ DOI 10.14712/1213DOI 10.14712/ -7243.2015 and ACG δ functions and showed that the classes of ACG * and ACG δ are equivalent on compact intervals.…”

Absolutely continuous variational measures of Mawhin’s type

“…At the same time, σ-finiteness of a variational measure gives some information about differentiability properties of the set function that determines this measure (see [2,4,6,7,10,13,21,24,32,35,36,40,41,42,44,49]). It is also well-known that, unlike general situation, the absolute continuity of a variational measure implies its σ-finiteness (see [3,4,6,7,9,12,14,15,17,19,27,31,33,36,39,40,44,46,47,49]). This relation between the absolute continuity and σ-finiteness motivated several authors to find characterizations of σ-finite variational measures.…”