An iterated limits theorem applied to the Henstock integral (Q1374567)

A general theorem for interchanging iterated limits with respect to directed sets is used to obtain two sets of conditions (C) which are necessary and sufficient for the following convergence statement: If \(\{f_k\}_{k=1}^{\infty}\) is a sequence of Henstock-Kurzweil integrable functions defined on \([a,b]\) and such that \(\{f_k\}_k\) converges pointwise to \(f\) on \([a,b]\) and (C) is true then \(f\) is Henstock-Kurzweil integrable on \([a,b]\) and \(\lim_{k\to \infty} \int_a^b f_k = \int_a^b f\). The form of the found conditions (C) was motivated by the notion of the \(\gamma\)-convergence due to \textit{R. G. Bartle} [Real Anal. Exch. 20, No. 1, 119-124 (1995; Zbl 0828.26006)].