The descriptive definitions and properties of the AP integral and their application to the problem of controlled convergence (Q1322614)

For each definition of an integral, there is often an optimal version of convergence theorem. The authors proved such a theorem for the approximately continuous integral of Burkill, following the controlled convergence theorem for the Henstock integral as given by the reviewer [Lanzhou lectures on Henstock integration (1989; Zbl 0699.26004)]. Furthermore, the authors showed that the descriptive definition of the \(S\)-Henstock integral by \textit{R. A. Gordon} [Real Anal. Exch. 16, No. 1, 154-168 (1991; Zbl 0723.26005)] is equivalent to that of the Burkill integral as given by Ma and the reviewer.