Rating: 
Summary: terrific reference book
Review: I'm surprised to see some negative reviews for this book. I've long found it to be an extremely useful reference book and probably look more stuff up in this book than any other math book I own. Anyone who works with real analysis but is not an expert on the topic would find this useful.
Rating:
Summary: A Standard Text
Review: If you are going to study mathematics at the graduate level, your analysis book will likely be or have been Royden. The subject topics include the L.O.M, generalized measure theory, convergence theorems and the Radon-Nikodym (all in one semester).
Rating: 
Summary: Why Everyone Use It for Textbook?
Review: It fascinates me everyone use this old book for textbook again and again. It seems to me that if professors do not use it, they will feel being perceived by others as lesser classic. Well, no doubt about it, this is indeed a good textbook. I will use it for my class. I don't have to pre-study it before my class starts. OK, OK, I am lazy and admit that there is no major progress in real analysis needs to be taught or added. Damn, it is true for graduate school too! If you want change, go to EE or CS. Our math professors are quite comfortable with our oldies - Royden, Rudin, Spivak, Yoshida, Apostol,....maybe Lang, Strange,..... Hey, Let's come back to "Real" analysis, how about Aliprantis' new 3rd edition? Shi..., we don't like it. It has solution book! Have you ever heard the old saying in our department meeting - "To learn math students have to do all the hard problems and we... demonstrate the easy ones!" No wonder all the textbooks are so familar. Ya, this is another reason for not changing the textbook. The same definitions (of course, dummy!), the same theorems, the same examples and counter-examples, hey, the same problems! Who want new problems? Do you have the answer? Rudin's Mathematical Analysis was a great try with the introduction of "papers". What happen afterwards? No one dare to follow. Nonetheless, everyone is still yawning "Heeee...It is a great textbook! Mmmmm..." Geeeez, so much for the math textbooks!
Rating: 
Summary: Pretty good as a first book, except for chapter 5.
Review: Royden is pretty good for learning about measure theory for the first time. There are some annoying misprints in the problems which cause headaches for students. A major wart is that Chapter 5 on differentiation is terrible. He keeps applying the vitali lemma over and over again, confusing the reader because he neglects to even mention Lebesgue points.
Rating:
Summary: The standard by which analysis texts are judged.
Review: Royden's text begins with a careful development of Lesbesgue measure and integration, with a discussion of differentiation and the L^p spaces. The book also provides a good introduction to metric spaces, including a discussion of Banach and Hilbert spaces. Also included is a very insightful and user friendly introduction to topological spaces. The later portions of the text present a well written development of abstract measure theory including signed measures, the Radon-Nikodym Theorem, Fubini and Tonelli Theorems, and the Riesz Lemma. Overall, the book is an indispensible tool for serious mathematics students. It is a very readable introduction to ideas central to mathematics as well as an invaluable reference.
Rating:
Summary: IF YOU DON'T HAVE TO BUY IT, DON'T BUY IT.
Review: Strictly, from a student's perspective this is not a good book, even though some professors might like it. First, this book have numerous misprints, especially in exercises, so you might often find yourself trying to prove something which is false and, in this case, it is really a waste of time. Second, the 1st section of a book, develops measure theory for a real line, while the 3rd -- for general measure spaces. Now, this is a real waste, as the general theory, basically consists of the same theorems, so there is not point in proving everything twice.
Rating:
Summary: This is a pretty average Real Analysis book
Review: The book presents a nice introduction to Lebesgue integration theory. There are a lot of typos so it isn't a good book to learn on your own from because his notation is not rigors. The book doesn't make a good reference either, being that a lot of the important results are left as exercises. So I would have to say this book isn't great, but I would recommend it over a lot of the other analysis books out there. I personally found that Halmos "Measure Theory," was pretty good.
Rating:
Summary: This is a pretty average Real Analysis book
Review: The book presents a nice introduction to Lebesgue integration theory. There are a lot of typos so it isn't a good book to learn on your own from because his notation is not rigors. The book doesn't make a good reference either, being that a lot of the important results are left as exercises. So I would have to say this book isn't great, but I would recommend it over a lot of the other analysis books out there. I personally found that Halmos "Measure Theory," was pretty good.
Rating:
Summary: Excellent Reference, Poor Introduction
Review: This book is an excellent reference for analysis. The proofs are modern and generally excellent, and the treatment of measure and integration is very advanced. However, I would not recommend this book as an introduction; not very much motivation or intuition is provided for the material, and the double-coverage of abstract and real cases is awkward. Recommended substitute: Walter Rudin, Real and Complex Analysis, and Kolmogorov, Introductory Real Analysis.
Rating:
Summary: Why to write two books and bound them as one?
Review: This is a very curious book. It performs the construction of the same integration theory twice! I still wonder why. In the first part the author builds up the theory for the real-variable case, and in the second for the general Banach-space case. The first one is totally included in the second one, so, why to do it? Perhaps to make the book thicker and more expensive? It would be enough to point out the real-variable case as an observation and even as an excercise. None of my teachers used it as textbook and I have used it only as a reference, but not the most important of them.Other features I don't like are that some interesting results are left for the reader as excercises, and that most of its notations are not standard. But nevertheless the text itself is not so bad. Please check my other reviews in my member page (just click on my name above).
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