CHARACTERIZATIONS OF AC*G∩C, AC¯*G∩Ci, AC AND AC¯ FUNCTIONS

Abstract

In connection with the study of $A{C}^{*}G$ functions, Lee Peng Yee introduced a condition which lies somewhere between $AC$ and Lusin’s condition $(\text{N})$, and it is called the strong Lusin condition. This condition also appears in Bordaon’s Lemma 2 of [4]. (In [7] Lee and Vyborny mentioned that this condition was also studied by Kurzweil, Jarnik and Schwabik.) Denoting this condition by $Y_{D^0}$, we show that $Y_{D^0}=A{C}^{*}G\cap \mathcal{C}$ on a closed interval.

There are also given several characterizations for the classes ${\underset{¯}{AC}}^{*}G\cap {\mathcal{C}}_i$, $AC$ and $\underset{¯}{AC}$. For these tasks we have developed a study of various interesting conditions, such as: $VB$, $\underset{¯}{VB}$, $V{B}^{*}$, ${\underset{¯}{VB}}^{*}$, $AC$, $\underset{¯}{AC}$, $A{C}^{*}$, ${\underset{¯}{AC}}^{*}$, $A{C}^{**}$, ${\underset{¯}{AC}}^{**}$, $A{C}_{D^{\#}}$, ${\underset{¯}{AC}}_{D^{\#}}$, $A{C}_{D^0}$, ${\underset{¯}{AC}}_{D^0}$, $A{C}_D$, ${\underset{¯}{AC}}_D$, $Y_{D^{\#}}$, ${\underset{¯}{Y}}_{D^{\#}}$, $Y_{D^0}$, ${\underset{¯}{Y}}_{D^0}$, $Y_D$, ${\underset{¯}{Y}}_D$ ($A{C}_{D^0}$ and $A{C}_D$ were introduced by Gordon in [4]).