Most Helpful Customer Reviews
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21 of 22 people found the following review helpful:
4.0 out of 5 stars
Excellent Moore Method Intro Text, December 19, 2001
I used this text as senior undergraduate in an introductory course to real variables. The course was structured as a sort of modified RL Moore method class: there were very few lectures, and we (the students) could only use theorems and propositions presented in the text if we had gone to the board and presented a valid proof for each. As such, most of the students learned the fundamentals very well. This in turn made my first graduate course in real variables much easier.The biggest downside however is that most graduate students don't have the time needed to dedicate to the various problems in this text, which is why Royden is probably not the best choice for a first year graduate text. Instead I would recommend Bartle's Elements of Integration and Lebesgue Measure as a first year grad text on the subject. It was disappointing to use Bartle and discover that so many of the problems in Royden, which I had spent countless hours attempting to prove, had been completely worked out in Elements of Integration. In short, Royden makes you work for many (most?) important results, and in the long run this makes for a much stronger understanding of the material- if you have the time to devote to it.
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26 of 30 people found the following review helpful:
4.0 out of 5 stars
Not perfect, but better than the rest, August 22, 2005
I'm a PhD student in mathematics at Georgia Tech. I used this when I first took graduate real analysis at North Dakota State, and then used Wheeden and Zygmund's Measure and Integral here at Georgia Tech as well as Folland's Real Analysis when studying for comps. Time and time again, I found myself going back to Royden for his well-written expositions that left enough out to keep you paying attention but wasn't so sparse that you couldn't figure out what was going on. Some here have complained about it doing everything twice. This can be a problem in some cases, such as common texts for a first course in real analysis where topological ideas are covered for Euclidean space first and then again for general metric spaces, but with measure theory, this is the right approach. I saw it first hand last fall, as my colleagues in another section treated Lebesgue measure on the real line as a special case and did things in generality, while my section dealt with R^n first and then moved on to general measures. In the end, I'm quite sure the section that I was in had a firmer grasp on the material.
Royden's classic work has withstood the test of time, and deserves to remain a standard text for years to come.
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13 of 14 people found the following review helpful:
5.0 out of 5 stars
Classic text on measure & integration theory, August 22, 2006
Many people criticize this book as unclear and unnecessarily abstract, but I think these comments are more appropriately directed at the subject than at this book and its particular presentation. I find this classic to be one of the best books on measure theory and Lebesgue integration, a difficult and very abstract topic. Royden provides strong motivatation for the material, and he helps the reader to develop good intuition. I find the proofs and equations exceptionally easy to follow; they are concise but they do not omit as many details as some authors (i.e. Rudin). Royden makes excellent use of notation, choosing to use it when it clarifies and no more--leaving explanations in words when they are clearer. The index and table of notation are excellent and contribute to this book's usefulness as a reference.
The construction of Lebesgue measure and development of Lebesgue integration is very clear. Exercises are integrated into the text and are rather straightforward and not particularly difficult. It is necessary to work the problems, however, to get a full understanding of the material. There are not many exercises but they often contain crucial concepts and results.
This book contains a lot of background material that most readers will either know already or find in other books, but often the material is presented with an eye towards measure and integration theory. The first two chapters are concise review of set theory and the structure of the real line, but they emphasize different sorts of points from what one would encounter in a basic advanced calculus book. Similarly, the material on abstract spaces leads naturally into the abstract development of measure and integration theory.
This book would be an excellent textbook for a course, and I think it would be suitable for self-study as well. Reading and understanding this book, and working most of the problems is not an unreachable goal as it is with many books at this level. This book does require a strong background, however. Due to the difficult nature of the material I think it would be unwise to try to learn this stuff without a strong background in analysis or advanced calculus. A student finding all this book too difficult, or wanting a slower approach, might want to examine the book "An Introduction to Measure and Integration" by Inder K. Rana, but be warned: read my review of that book before getting it.
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