Some difficulties in education of Lebesgue integral are connected with
necessity to built measure theory. Of course, there are approaches using only intervals
(e.g. , , ). Our approach is inspired by the textbook . In the first chapter of
the book there is built the theory of real numbers, it means their construction. Of course,
the author of the book recommends to miss this chapter and to work from the beginning
with real numbers and their properties. Analogously we do not construct measure of
some complicated sets, but we assume that it is given.