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:
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.