On the equivalence of McShane and Lebesgue integrals (Q1374565)

It is known that a given function \(f:[a,b]\to \mathbb R\) is Lebesgue integrable on \([a,b]\) if and only if it is McShane integrable. (For the definition of the McShane integral, which is a sum type integral related to the Henstock-Kurzweil integral [see, e.g., Definition 16.7 in \textit{P.-Y. Lee}, ``Lanzhou lectures on Henstock integration'' (1989; Zbl 0699.26004)] The author gives a new proof of this result which does not rely on measure theory. In particular, he does not make use of the Egoroff and Luzin theorems.