The \(S\)-Henstock integration and the approximately strong Lusin condition (Q1322649)

In recent years work has been done to extend the Henstock-Kurzweil integral and associated concepts to integrals related to the approximate derivative. In particular, \textit{K. Liao} and \textit{T.-S. Chew} [Real Anal. Exch. 19, No. 1, 81-97 (1994)] have extended the strong Luzin condition of \textit{P.-Y. Lee} [Real. Anal. Exch. 15, No. 2, 754-759 (1990; Zbl 0716.26004)]. In this paper this extension is used to define an integral that is shown to be equivalent to a Henstock-Kurzweil integral introduced by \textit{R. A. Gordon} [Real. Anal. Exch. 16, No. 1, 154-168 (1991; Zbl 0723.26005)]. In particular, the author proves that the approximate strong Luzin condition of \textit{Liao} and \textit{Chew} implies the classical Luzin condition (N), answering a problem posed by Lee.