Characterizations of \(\mathbf{VB^*G} \cap \mathbf{(N)}\)

Abstract

We introduce the condition \((PAC^*)\) that is a slight modification of the condition \((PAC)\) of Sarkhel and Kar. The main result is Theorem 4: \emph{A function \(f:[a,b] \to {\mathbb R}\) is \(VB^*G ąp (N)\) on a subset \(E\) of \([a,b]\) if and only if \(f \in (PAC^*)\) on \(E\)}. Consequently, \emph{the set \(\{f:[a,b] \to {\mathbb R}: f \in VB^*G ąp (N)\) on \(E\}\) is an algebra, whenever \(E\) is a subset of \([a,b]\)}. Using Theorem 1, we find seven characterizations of \(VB^*G ąp (N)\) on a Lebesgue measurable set (Theorem 5). We also give fifteen characterizations of the class of \(AC^*G\) functions on a closed set \(E\), that are continuous at each point of \(E\) (Theorem 6). In the last two sections, using Thomson’s outer measure \({\mathcal S}_o\text{-}\mu_f\), we characterize a \(VB^*G ąp (N)\) function \(f\) on a Lebesgue measurable set (Theorem 9). As a consequence we obtain that: \emph{A function \(f:[a,b] \to {\mathbb R}\) is \(AC^*G\) on a closed subset \(E\) of \([a,b]\) and continuous at each point of \(E\) if and only if \({\mathcal S}_o\text{-}\mu_f(Z) = 0\) whenever \(Z\) is a null subset of \(E\)} (Theorem 10).