On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals. (Q2810141)

Let \(E\subset\mathbb R^n\) be a closed nondegenerate interval and let \(\{f_{n}\}\) be a sequence of Kurzweil-Henstock integrable real-valued functions on \(E\). For each \(n\), let \(F_n\) denote an additive interval function defined by NEWLINE\[NEWLINEF_n(I)=\int_If_nNEWLINE\]NEWLINE for each closed nondegenerate interval \(I\) in \(E\). Under this notation, recall that{\parindent=0.7cmNEWLINE\begin{itemize}\item[(i)] \(\{f_n\}\) is said to be \textit{equiintegrable} on \(E\) if for each \(\varepsilon>0\) there is a gauge \(\delta\) on \(E\) such that, for all \(n\), NEWLINE\[NEWLINE(D)\sum| f_n (x)| I| -F_{n}(I)| <\varepsilonNEWLINE\]NEWLINE whenever \(D\) is a \(\delta\)-fine partial division of \(E\);NEWLINE\item[(ii)] \(\{f_{n}\}\) is said to satisfy the \textit{uniform double Lusin condition} (or \(\mathrm{UI}_1\), in short) on \(E\) if for each \(\varepsilon>0\) there is a gauge \(\delta\) on \(E\) such that, for all \(n\), NEWLINE\[NEWLINE(D)\sum| f_n(x)| | I| <\varepsilon\text{ and }(D)\sum| F_n(I)| <\varepsilonNEWLINE\]NEWLINE whenever \(D\) is a \(\delta\)-fine partial division of \(E\) in \(\Gamma_{\varepsilon,n}\), where NEWLINE\[NEWLINE\Gamma_{\varepsilon,n}=\{(x,I):I\subset E, x\text{ is a vertex of }E\text{ and }| F_n(I)-f_n(x)| I| | \geqslant\varepsilon| I| \};NEWLINE\]NEWLINE \item[(iii)] \(\{f_n\}\) is said to satisfy the \(\mathrm{UI}_2\) \textit{condition} if for each \(\varepsilon>0\) there is a gauge \(\delta\) on \(E\) such that, for all \(n\), NEWLINE\[NEWLINE(D)\sum| I| <\varepsilon\text{ and }(D)\sum| F_n(I)| <\varepsilonNEWLINE\]NEWLINE whenever \(D\) is a \(\delta\)-fine partial division of \(E\) in \(\Gamma_{\varepsilon,n}\).NEWLINENEWLINE\end{itemize}}NEWLINEThe key result of the paper under review, Theorem 3.3, compares the above three notions:NEWLINENEWLINENEWLINESuppose that \(f_n\rightarrow f\) pointwise on \(E\). Then, each of the above three statements about \(\{f_n\}\) implies all the others, Kurzweil-Henstock integrability of the function \(f\) on \(E\), and the relation NEWLINE\[NEWLINE\int_Ef=\lim_{n\rightarrow\infty}\int_Ef_n.NEWLINE\]NEWLINE In the remainder of the paper, the authors extend this result to the context of general division spaces.