Definition of Lebesgue Integral, why define integral for bounded functions? [Stein] 
Stein's development for integral:

(1.) Define the integral of simple measurable functions. 
(1'). Define the integral of bounded measurable functions on sets of finite measure with (1.) 
(2.) Define the integral of nonnegative measurable functions with (1'.) 
My question: I do not understand why 1' was necessary in Stein's development, what additional insight does it give? 
 A: Firs of all, this is more like a taste from the author than a need. But you can justify this choice. There is some possible reasons:
First - Develop some theory in an simple background smooth the latter complexification.
Second - A lot of the proofs for important theorems in abstract background are identical to the proofs for the weaker version of these theorems applied to small classes of functions. So you can either turn simple for the reader to understand or leave for the reader to write down the demonstration in the actual more general background. (For example, the proof of Proposition 1.6(vi), Chapter 2 of Stein's book is left to the reader this way)
Third - Time to time you will have to solve a problem for a function or a sequence of functions (for example) that, if you approximate by more basics functions, the problem turns easy.
I like the way Rudin writes his books, and don't think that this type of approach is more useful than that of Rudin (although these justifications). However, new books have to differ from the old. Stein's book is new in comparison, for example, with Rudin's book, so maybe the author just wanted to do things differently.
