Uniform generalized absolute continuity and some related problems in \(H\)- integrability (Q1322646)

The concept of absolute continuity needed to describe the indefinite Henstock-Kurzweil (= Denjoy-Perron) integral has been given several different but equivalent forms. These in turn have been used to prove limit theorems, controlled convergence theorems, by using the associated generalization of uniform absolute continuity that originally appeared in the classical result of Vitali [this theorem and the original extension to the more general case can be found in ``The theory of the Denjoy integral and some applications'' by \textit{V. G. Chelidze} and \textit{A. G. Dzharshejshvili} (Russian original 1978; Zbl 0471.26005; English translation 1989; Zbl 0744.26006)]. In this very interesting paper the authors consider six classes of uniform generalized absolute continuity arising from six equivalent forms of absolute continuity. The interesting result is that not all the uniform classes are equivalent. Four are: let us call this class \(\text{UACG}^*\), and the remaining two \(\text{UACG}_{**}\), \(\text{UACG}_ *\). Then the authors show that: (i) \(\text{UACG}_{**}\subset\text{UACG}_ *\), by constructing an example that gives the strict inclusion; (ii) \(\text{UACG}_{**}\subseteq \text{UACG}^*\), leaving open the question as to whether this inclusion is strict. This also leaves open the relation between \(\text{UACG}^*\) and \(\text{UACG}_ *\). The rest of the paper is a discussion of various controlled convergence results answering problems posed by \textit{P.-Y. Lee} [Lect. Notes Math. 1419 (1990; Zbl 0701.00010)] and by \textit{R. A. Gordon} [Real Anal. Exch. 17, No. 2, 789-795 (1992; Zbl 0764.26002)].