# 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:

1. 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.

2. 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.

3. 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.

4. 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.
5. 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.

• Try Bass's Real Analysis for Graduate Students. Royden is also good. There's a real question here, though, about how long you want to learn "real analysis" as opposed to moving on to other topics. (Evans PDE textbook seems to assume relatively little real analysis, in fact.) You can always come back to things (e.g. Riesz rep theorem, measure version) that you want to understand better. – fourierwho Jun 29 '18 at 21:55
• @fourierwho You're right. I think everyone definitely goes back and forth the material, and the relatively advanced topics/exercises make better sense only after some time. However, given that I'll be taking a course in the fall, I'd like to learn the subject in as much detail as I can in a semester. After the semester ends, I would like to go on to the PDE's/Probability part. Suggestions? – user82261 Jun 29 '18 at 22:09
• @fourierwho Also, are you referring to Royden and Fitzpatrick's book (4th edition)? Or to a previous edition. Wouldn't it take a painstakingly long time to cover material from Royden? I mean, the 4th edition spends 150-200 pages covering everything in $\mathbb{R}$ first. I'm not sure if it'd be a good idea, given that Rudin and Royden will take completely different approaches. I think the reference book for the course is Stein and Shakarchi's Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Thoughts on it? – user82261 Jun 29 '18 at 22:10
• Yes, Royden and Fitzpatrick. I agree you should only read word-for-word a small fraction of Royden. Same could be said for any textbook (in a reasonable period of time). Anyway, I'll write an answer. – fourierwho Jun 29 '18 at 22:21
• @fourierwho All right. I'll be looking forward to read your answer. Please do comment on Stein and Shkarchi's textbook as well. From what I gather, this textbook also covers Lebesgue integration on the real line first, before moving on to abstract measure theory. Is it a better choice than Royden's book? – user82261 Jun 29 '18 at 22:26

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.

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.)

• Thanks for the detailed response. Ahh, I don't think I may have enough time to cover all these things prior to the start of the term. Since right now, I'm trying to revise for the analysis and algebra qualifying exams upon arrival. The good thing, however, is that I did cover a bit of measure theory by auditing a course from Bartle's Elements of Integration and Lebesgue Measure Theory." The good thing is that the textbook covers abstract measure theory first. But, since I was auditing the course, I didn't do everything in detail. For example, I didn't cover Fubini's Theorem in any detail. – user82261 Jun 29 '18 at 22:39
• Also, naturally, all the details that I covered 6-8 months ago are now a bit hazy. For example, the construction of Borel measures (?) etc. is something that I have completely forgotten. I do, however, remember the basics: how the integral is defined, convergence theorems etc. The book didn't have a lot of questions, so that's why these details are a bit here and there. I am able to retain material only when I am able to solve a large number of problems. Given this, what resource should I keep on the side in case I'm not able to find the time go through your aforementioned plan over – user82261 Jun 29 '18 at 22:43
• the next month. – user82261 Jun 29 '18 at 22:43
• Bass is a good reference, both for reading and problems. Same is true of Royden. – fourierwho Jun 29 '18 at 22:46
• Right. But circling back, wouldn't you think it would not make a lot of sense to keep Royden's book (and even Stein's) on the side with Rudin'a book, since both the books take radically different approaches. I was perhaps whether one can find a book that strikes a reasonable middle ground. If you have any other suggestion, please do share. Your initial response was quite helpful, though. – user82261 Jun 29 '18 at 22:53