On Saks-Henstock lemma for the Riemann-type integrals (Q1194683)

A function \(f\) is \(RL\) integrable on \([a,b]\) [for the definition see the paper of \textit{Ding Chuan-Song} and \textit{Lee Peng-Yee}, Real Anal. Exch. 12(1986/87), 524-529 (1987; Zbl 0659.26004)] if and only if there exists an absolutely continuous function \(F\) such that for every \(\varepsilon>0\) and \(\eta>0\) there exist an open set \(G\) and a constant \(\delta>0\) such that \(| G|<\eta\) and that for every division \(D=\{([u,v],\xi)\}\) with \(0<v-u<\delta\) and \(\xi\in[u,v]\) we have (D) \(\sum_{\xi\notin G}| f(\xi)(v-u)-(F(v)-F(u))|<\varepsilon\). This theorem is used to prove a mean convergence theorem.