# Ways of learning Lebesgue integration

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.

mathwonk
Homework Helper
2020 Award
i think you are losing nothing. the integral is treated in the way you describe in several classic, excellent texts including the one you are using.

others include the great Functional Analysis, by Riesz and Nagy, and the fine Analysis II by Lang.

There is a simple interplay between sets and functions ni which a set is mirrored by its characteristic function, the fu ntion whiuch equals 1 on the set and 0 off it. The set is measurable iff the function is measurable.

thus approximating functions by step functions is like aproximating sets by rectangles.

one can begin by defining integrals of step functionsm and taking limits of step functions nd studying when the limits of step functions have integrals, or one can begin by studying sets which are limits of rectangles and when etc etc.....

if you have measure theory first then the integral of a positive fucnion can be defined as the measure of the region under the rgaph, and if you have integration first then the measure of a set is the integral of its characteristic function.

comme ci comme ca.

I am not an expert however, indeed a rookie (an old rookie) of sorts in this area.

Thanks, mathwonk. I am studying by myself and sometimes I need this kind of information to reassure me in what I am doing.