A simplified construction of the Lebesgue integral (Q2322495)

Since 1902, Lebesgue's introduction of an integral, now called the Lebesgue integral, there appeared several alternative proofs for the related constructions and results (mostly convergence theorems). Probably the shortest and most elementary approaches to the Lebesgue integral is due to Riesz from 1949. Among recent publications devoted to the Lebesgue integral, recall \textit{W. Johnston} [The Lebesgue integral for undergraduates. Washington, DC: Mathematical Association of America (MAA) (2015; Zbl 1331.26001)]. Now, this paper of V. Komornik is a further step forward in these simplifications. Komornik's elegant approach deals with larger classes of functions than Riesz (finiteness of integrals is relaxed in some of his steps). After introducing, by means of two fundamental Lemmas, the Lebesgue integral on the real line, dealing with the length of intervals, i.e., dealing with the standard Lebesgue measure, the general Lebesgue integral acting on an arbitrary measure space, and, in particular, for product measures, is given. To stress the differences between the author's and Riesz's approaches, disappearing of various difficulties and counter-examples related to the author's simplified approach are explained if one return to the original Riesz's approach. The paper is transparent and readable also for beginners in measure and integral theory. Several results with proofs identical to those of the classical case are omitted in the main text, however, for the convenience of readers, they are added in an appendix.