Characterizations of \(\mathbf{VBG ąp (N)}\)

Abstract

We show that \(VBG \cap (N)\) is equivalent with Sarkhel and Kar’s class \((PAC)G\) on an arbitrary real set. Hence \(VBG \cap (N)\) is an algebra on that set. In Theorem 4, we give three characterizations for \(VBG \cap (N)\) on an arbitrary real set. It follows that Gordon’s \(AK_N\)-integral is a special case of the \(PD\)-integral of Sarkhel and De (Remark 3). In Theorem 3 we obtain the following surprising result: a Lebesgue measurable function \(f\) is \(VBG\) on \(E\) if and only if \(f\) is \(VBG\) on any null subset of \(E\). We also find seven characterizations of \(VBG ąp (N)\) for Lebesgue measurable functions (see Theorem 5). For continuous functions on a closed set, we obtain several characterizations of the class \(ACG\). Using a different technique, we obtain other characterizations of \(VBG \cap (N)\) for a Lebesgue measurable function (see Theorem 8).