My friends who are expert analysts liked Wheeden and Zygmund. I who am not, always liked as a student, everything by Sterling K. Berberian.
Be aware there are several ways to do "Lebesgue integration", namely with and without measure theory. In my own opinion, treatments that do not use measure theory may sometimes be easier, but may leave you ignorant of what some feel is a basic topic.
Berberian's books do cover measure theory as I believe also does Wheeden and Zygmund. A book that does not is the classic by Riesz and Nagy. A book that makes a hybrid of the two points of view is Real Analysis by Lang. Lang's book can be very abstract and overwhelmingly general.The essential difference is this: It is possible for a sequence of functions whose Riemann integrals exist to converge pointwise to a function whose Riemann integral does not exist, even though the sequence of integrals does converge.
There are two solutions to this problem: the usual, measure theory way is to start all over with a new definition of the integral using hard measure theory for complicated sets much more complicated than the intervals that Riemann's theory is based on, and show that in this new theory the limit function does have an integral which does equal the limit of the integrals of the Riemann integrable functions.
Another way, to me much easier is to simply say that the limit of any function that arises as a pointwise limit (almost everywhere) of a sequence of Riemann integrable functions with convergent integrals, is integrable and its integral is that limit. Then one has to prove that this limit is independent of the choice of Riemann integrable functions converging to the given limit function. I find this approach much simpler.
In both approaches one wants the result that the Lebesgue integrable functions form a complete metric space, in the L1 (i.e. "integral") norm, in which the Riemann integrable functions are dense. I admit that in my preferred "simpler" approach, one has to prove then that every L1 Cauchy sequence of functions does converge almost everywhere pointwise to some function, the limit function.
Purists however may feel that Lebesgue measure is so beautiful that it deserves to be developed on its own. Ok. Just be aware that different books are trying to serve different purposes of their own, not necessarily to get you where you want to go as quickly and easily as possible.Basically, no matter how you slice it, you need to prove that every L1 Cauchy sequence of integrable functions has a pointwise limit almost everywhere, and that for any two L1 Cauchy sequence sequences with the same pointwise limit a.e., their integrals converge to the same limit.
Then you want a fundamental theorem of calculus, basically that an integrable function g is the derivative a.e. of its indefinite integral, and a function F is an indefinite integral (necessarily of its derivative) if and only if F is absolutely continuous.
This is my take on a subject I am not at all expert in and have always been puzzled by, so run these opinions past an actual analyst in a university, before betting on them.
