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I'm taking a course in measure theory and the professor isn't following a textbook. Of course, he will be assigning homework problems, but I feel like just doing homework problems is never enough for me to fully understand the material. My plan is to have a separate "problem book" in addition to my class, which I'll keep on solving as we keep on studying new topics in class (i.e. I'll solve the relevant exercises after we cover the material in class, and keep this going continuously). I have three options for books to make my problem book:

  1. Real Analysis by Royden
  2. Stein and Shakarachi's Measure Theory notes
  3. Rudin's Real and Complex Analysis

Which one is best as a "problem book"? A bit more on what I want from this book: I want exercises to be able to do quickly as I study the material. I'm sure the homework will have enough "deep problems", so my goal here isn't to find a book with interesting problems, but one that can serve as a book to make me really get warmed up with the material and understand it better before attempting homework problems. Plus, I don't have that much extra time to spend on these problems, so I'm not looking for the hardest problems either. Which one of these is best for this purpose? I've used Royden quite a lot before, but don't like its presentation of measure theory too much (they break up measure theory into two sections: one just focusing on $\mathbb{R}$ and one in an abstract setting, and sort of repeat the same thing twice, though I still like the book), and don't have much experience with Stein and Rudin.

PS - perhaps relevant to this post, but I've studied measure theory before, so it's not my first time seeing most of these things.

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  • $\begingroup$ real analysis by carothers. rudin's real and complex analysis is definitely not what you want (the problems are not easy). $\endgroup$ Oct 7 '20 at 11:22
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You might want to take a look at my book Measure, Integration & Real Analysis, which was published earlier this year in Springer's Graduate Texts in Mathematics series. I placed this book in Springer's Open Access program, which means that the electronic version of the book is legally available for free at http://measure.axler.net/.

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I think Bartle's Elements of Integration and Lebesgue Measure might be the book for you. It's problems are not extremely difficult, but they are interesting nevertheless, and overall I think it is a good book.

When I took measure theory I used it alongside Folland's Real Analysis modern techniques and their applications, and I think it was a great match in terms of difficulty of the exercises and explanation of the subject.

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Not a book but the Measure Theory for statistics course at Duke has great exercises (and great content too): http://www2.stat.duke.edu/courses/Fall20/sta711/

The homework problems are mostly results you have to prove and the included past midterm and final problems are more of a speed-thinking/applied variety.

All in all its a great resource

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You can check out Measures, Integrals and Martingales 2nd Edition by René L. Schilling. This is an introductory book though, but you mentioned that you need warm up. Moreover, all the exercise have detailed solutions on author's webpage.

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