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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?

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    I think that it is only a "pedagogical" choice : going step-by-step can make exposition more smooth. Compare with Sergei Ovchinnikov, [Measure, Integral, Derivative : A Course on Lebesgue's Theory](https://books.google.it/books?id=bt69BAAAQBAJ&pg=PA65) (2013), page 65. – 2017-02-05
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    As you can see, there are different "staged" approaches : 4 (Stein), 3 (Ovchinnikov), 2 (Rudin). – 2017-02-05
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    Hmm May you elaborate on how it make the explanation easier? I find Rudin's simplest... – 2017-02-05
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    Frankly speaking, I've no teaching experience... My perosnal "feeling" is that the motiviations under the different expositions are only pedagogical. – 2017-02-05

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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.