Good day. I am studying Lebesgue integration in Apostol’s Mathematical Analysis. I have learned already (I believe so) the Dominated Convergence Theorem and the Theorem of Differentiation under the integral sign. But Apostol does not introduce the Lebesgue integration by way of a Theory of Measure. Instead he prefers to define Lebesgue integrals in this order: for step functions, for upper functions, for Lebesgue functions, for measurable functions. (F is a measurable function in an interval I if and only if it is the limit of a sequence of step functions). Does someone of you have learned Lebesgue integration in this way? Is it equivalent to a study based directly on Measure Theory? Do I am losing something relevant? Thanks for opinions and excuse the mediocre english.