Supplement to Rudin's Real and Complex Analysis I will hopefully be taking a course on real analysis taught from Walter Rudin's Real and Complex Analysis this fall. I'm looking for a supplement book that meets (at least some of) the following criteria:


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*As per this book review and some comments made at the start of chapter 2 of the book, Rudin seems to prove almost everything in the most general setting. For example, I think the Riesz representation theorem is proved for locally compact (Hausdorff?) spaces. I would like a book that presents the same material at the same level or at a lower level (in a specialized setting), making it easier for the reader to understand the concepts.

*The textbook should contain both proofs and concrete examples. For example, I wouldn't mind if the book has a large number of examples of complicated looking integrals that can otherwise be solved rather easily using convergence theorems.

*The book should contain examples and counter-examples. I have observed that a lot of questions in Rudin's books are about constructing examples/counter-examples. Whereas the textbook, on the other hand, does very little on this front. For example, I'm aware that the Cantor Function is used as a counter-example/example to illustrate some measure-theoretic concepts in $\mathbb{R}^1$, and this is something that is explained in detail in, say, Royden's book. I such at coming up with/remembering examples/counter-examples, so I'd like a book that meets this criteria.

*The textbook should have more approachable exercises. I found that I enjoyed studying Rudin's PMA after having studied analysis from other books. I came to realize that the exercises are non-trivial, and each exercise has a meaning of its. So, Rudin's PMA starting making sense me only in hindsight. I think I'll have the same experience with Rudin's RCA, so I'd like a book that has more approachable, and a large number of exercises.

*I'd like to start studying PDE's and Probability after studying real analysis.


It'd be great if someone could recommend a supplementary book. 
 A: Based on the comments, I would say you could consider doing a few things before your course:
1)  Read Chapter 2 of Royden-Fitzpatrick
2) Read the first chapter of Friedman's Foundations of Modern Analysis, up to and including section 1.6.  A very elegant approach to abstract measure theory with a number of concepts (i.e. limsups of sets) that are usually neglected nowadays.  Also, a much more gentle intro to abstract measure theory than Rudin (e.g. no complex numbers to deal with).  
3) Learn about measurable functions and Lebesgue integration from Bass Real Analysis for Graduate Students (Chapter 5, 6, 7) or Friedman's book (Chapter 2, up to and including Section 2.10).
4) If you're still alive at this point, try reading Chapters 4, 5, 6 of Royden.  (By alive, I just mean you may not get this far, and that's OK.)
You won't have seen the Riesz representation theorem up to this point, but you will know the basics of Lebesgue integration.  For reference, (1-3) is roughly what I did to learn basic measure theory the first time, and (4) was part of the grad course I took as an undergrad.  (I didn't take the grad course as a PhD student.)
Stein and Shakarchi (Measure Theory, Hilbert Spaces, etc.) would be similar to Chapters 2-6 of Royden, but its exercises are harder (and not in a good way).  The fact that it doesn't do "abstract" stuff until the end doesn't recommend it either.  (Incidentally, it has nice extra material on geometric measure theory at the end, but that's a different story.)
A: If you want a different viewpoint on measure theory and integration, and since you plan to study probability theory later anyway, one idea would be to study probability theory now in a book that develops the Lebesgue integral from scratch. For example, you could use Billingsley's Probability and Measure. It has good exercises, and probability theory provides lots of concrete examples.
There's a French book I haven't seen in a long while, but as I recall it was similar to Rudin though slower moving, less comprehensive, and with easier exercises. It's called Intégration, by André Gramain.
