The Wide Denjoy Integral as the Limit of a Sequence of Stepfunctions in a Suitable Convergence

Abstract

In this paper we shall prove that a function \(f:[a,b] \to \overline{{\mathbb R}}\) that is \({\mathcal D}\)--integrable on \([a,b]\) can be defined as the limit of a \({\mathcal D}\)-controlled convergent sequence of stepfunctions (see the second part of Theorem 2). In the last section we show that Ridder’s \(\alpha\)- and \(\beta\)-integrals can also be defined as the limit of some controlled convergent sequences of stepfunctions (see Theorem 4).