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:
- Real Analysis by Royden
- Stein and Shakarachi's Measure Theory notes
- 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.
Best Answer
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/.