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Real Analysis
Real Analysis, Fourth Edition, covers the basic material that every reader should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in mathematics and familiarity with the fundamental concepts of analysis. Classical theory of functions, including the classical Banach spaces; General topology and the theory of general Banach spaces; Abstract treatment of measure and integration. For all readers interested in real analysis.
349 pages, Hardcover
First published December 1, 1963
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April 2, 2021
Analysis on the real number line, such as one encounters in an introductory course at the advanced undergraduate level (using, say, Rudin’s Principles of Mathematical Analysis as a textbook), constitutes only a preliminary to a vast and far-reaching domain, the subject of real analysis properly so called. Hence, as a beginning graduate student, it is imperative to return to the subject and relearn it from the most advanced point of view. Royden’s Real Analysis, the subject of this review, has served for a long time to define the consensus curriculum (this was the default choice for upper-level undergraduate majors in mathematics at Princeton when the reviewer was in college there). Recently, the text by Folland has offered it stiff competition and begun to displace it; this reviewer will not comment very much on this development, as he lacks sufficient experience with the latter, other than to say that it seems to be a step more sophisticated in outlook in its compressed treatment of the standard material, and it goes on to applications, for instance to probability theory and to partial differential equations, that will not be found in Royden at all. Moreover, let it be remarked that we stay with the third edition of Royden’s Real Analysis. There is now a fourth edition out by another author, but, to this reviewer at least, it appears to be a travesty, an attempt to make Royden over into a completely different pedagogical model of how to present the subject. It is a mystery how an apprentice who thus violates Royden’s authorial intent could get his text posthumously recognized as a fourth edition of the master himself! There should be a statute of limitations.
Royden’s style is disarmingly simple. Unlike many authors, who dress things up in fancy notation to look more complicated than they really are, Royden’s prose is clear and lucid and to the point. After necessary preliminaries on logic and set theory in chapter one, perhaps Royden’s most controversial pedagogical decision is to present the elements of measure theory and Lebesgue integration first in the special case of the real line in Part I, chapters two through six. As a result, many arguments have to be recapitulated in Part III on measure and integration theory in general spaces. A Bourbakian purist would object to this manner of exposition, but here is this reviewer’s two cents’ worth: learning always proceeds from the concrete to the abstract, especially in a technical field such as mathematics. One does not fully understand a concept or its larger significance until one has worked his way through a number of non-trivial examples in detail and gained familiarity with how things behave from first-hand experience, not from being told so (this is why it is necessary to solve many homework exercises by oneself). Needless to say, the historical development by which the concepts themselves were originally discovered follows a similar pattern. The historian of science Thomas Kuhn draws attention to these elementary facts about learning in his theory of scientific revolutions with his idea of a ‘paradigm’, in its proper and most appropriate sense of an exemplary solution of a problem (not the loose sense in which the word is often bandied about these days, to mean a whole constellation of ideas or ‘Weltanschauung’ as in Einstein’s paradigm of relativistic physics). In fact, before Kuhn raised the term into prominence, it had a rather limited meaning in grammar; a paradigm is a schematic declension of a verb, such as the high-school student of Latin will have encountered. Now, logically of course measure and integration could be taught by starting with the most general case and some economy would thereby be gained, as one finds in, say, Folland and Rudin (‘papa Rudin’, that is). But from a pedagogical point of view the supposed advantage of such an approach must be largely illusory. Repetition is of the essence of learning, after all. This reviewer finds it helpful to see concepts such as outer measure introduced first on the real line, where one can visualize everything. Moreover, chapter five on differentiation and integration is specific to the real line; the fundamental theorem of calculus in the form it receives from the point of view of Lebesgue theory makes sense only in one variable anyway.
Let us return to reviewing Royden after this excursus. Chapter six on the classical Banach spaces (i.e., Lp spaces in the unit interval) is a good illustration of the material of real analysis when it goes beyond what the student will have seen in introductory calculus and analysis. Royden provides easy-to-follow proofs of the Minkowski inequality, Hölder inequality, Riesz-Fischer theorem (establishing completeness) and Riesz representation theorem for bounded linear functionals. At this point, the student should be ready to reproduce Royden’s argument in the case of the spaces of convergent sequences and of sequences convergent to zero.
Before taking up general measure theory in Part III, Royden launches into an exposition of abstract spaces in Part II. In part, this entails a review of familiar concepts on metric spaces and point-set topology for anyone who has gone through baby Rudin, but here they are pushed farther in sections on Baire category, the separation axioms, compact and locally compact spaces, the Stone-Čech compactification, the Stone-Weierstrass theorem for algebras of continuous functions in compact Hausdorff spaces, the closed graph and Hahn-Banach theorems, weak topologies and reflexivity and a little on topological vector spaces (through a somewhat abbreviated proof of the Krein-Milman theorem apparently taken from Kelley’s book on general topology). Some reviewers will object to Royden’s tendency to relegate what are substantial results in real analysis to the exercises, such as Urysohn’s lemma, Tietze’s extension theorem and the Urysohn metrization theorem. But a beginning graduate student should be able to handle them, and hints are provided anyway. The sections on topological vector spaces and Hilbert spaces are too rushed; they barely scratch the surface of what could be extensive fields in mathematics. There would be no space in a textbook on real analysis to cover such topics adequately, however, so perhaps one should be content with merely gaining some initial exposure to why these ideas matter for the sake of that elusive quality, mathematical maturity.
Given the extensive preparation the student will have received in Parts I and II, the presentation of abstract measure and integration in Part III can proceed at a fast clip. To alert the reader to what new phenomena can arise, compared with the case of standard Lebesgue measure on the real line, there are sections on signed measures, the Hahn decomposition theorem, the Radon-Nikodym theorem and the Lebesgue decomposition theorem. The standard results in general Lp spaces are recovered in the space of four pages. After this quick march through measure and integration theory in chapter eleven, chapter twelve considers methods of defining a general measure on a sigma-algebra, via outer measure and the Caratheodory extension theorem. Here, we also get sections on product measures and the Fubini and Tonelli theorems. The concepts are illustrated by means of a nice section on integral operators from Lq to Lp. The chapter is rounded out with sections on inner measure, Caratheodory outer measure and Hausdorff measure. The attentive student should take notice of what is being done here; what are standard results that one takes for granted in multivariable calculus, such as changing the order of integration, have to be established in the setting of general measure spaces.
This reviewer rather enjoyed the following chapter thirteen, in which Royden enters into some of the non-trivial complications that arise in locally compact Hausdorff spaces, with the concepts of Baire measure and regularity through the Riesz-Markov theorem for positive linear functionals on the space of continuous functions with compact support and the Riesz representation theorem for bounded linear functionals on the space of continuous real-valued functions on a compact Hausdorff space. The coverage in this chapter is extensive enough to get to some substantial results; the interested reader can find more in Halmos’ dated but classic text on measure theory.
In chapter fourteen, Royden applies the techniques developed so far to prove the existence of a Haar invariant measure on locally compact topological groups, making essential use of the Hahn-Banach theorem. The material on topological groups, group actions, unicity of the invariant measure and groups of diffeomorphisms comes across as too compressed to be really satisfactory, but it does illustrate some advanced applications of real analysis.
This reviewer fails to appreciate what chapter fifteen on mappings of measure spaces, Boolean sigma-algebras, measure algebras and Borel equivalence is all about. Suffice it to say that we get some further non-trivial applications of concepts in real analysis, through the Kuratowski theorem on Borel equivalence and a characterization of isometries on Lp[0,1]. The reviewer does appreciate the concluding chapter on the Daniell integral, though. The Daniell integral is a way of extending the usual Riemann integral to a larger class of functions without going through all the apparatus of measure and Lebesgue theory with the aid of a suitable positive linear functional which, nevertheless, enjoys all the desirable properties of the Lebesgue integral with respect to convergence of sequences of functions. The Daniell integral is probably not a canonical topic in real analysis and one could very well get along without it; still, it is nice to see it disclosed as a possibility.
Royden’s text is liberally supplied with homework exercises, some 620 in all over fourteen chapters. As a rule, they are either not too difficult or, in the case of results that might be more of a challenge to the beginning student, outfitted with a hint or plan of proof to follow. They do, however, illustrate the material nicely and it is a good idea to attempt many or most of them in order to solidify one’s acquaintance with real analysis.
If the reader is patient and sticks with it, he will end up with an exposure to real analysis at a somewhat advanced level, though not as far advanced as that of papa Rudin, say. The level attained, though, suffices to go on to functional analysis proper and the world of interesting applications it opens up (in mathematical physics, the theory of partial differential equations, probability theory, operator algebras etc.). The reviewer will hazard a surmise which, technically, he is not entitled to make, not having studied Folland’s competing text thoroughly: the apparently greater sophistication with which Folland treats the subject of real analysis is, for the most part, window-dressing and not really necessary to an adequate command of the material, which is to say, adequate for the important applications as opposed to a hypertrophic development merely for its own sake (one encounters a similar phenomenon in set theory, which most mathematicians need as basic material to use in their own disciplines but which a small band of devotees has extrapolated into a baroque monstrosity largely devoid of significant applications to real, everyday mathematics outside its own narrow domain). The reviewer does hope to work his way at least through the first half of Folland eventually, when time permits, so keep posted!
Royden’s style is disarmingly simple. Unlike many authors, who dress things up in fancy notation to look more complicated than they really are, Royden’s prose is clear and lucid and to the point. After necessary preliminaries on logic and set theory in chapter one, perhaps Royden’s most controversial pedagogical decision is to present the elements of measure theory and Lebesgue integration first in the special case of the real line in Part I, chapters two through six. As a result, many arguments have to be recapitulated in Part III on measure and integration theory in general spaces. A Bourbakian purist would object to this manner of exposition, but here is this reviewer’s two cents’ worth: learning always proceeds from the concrete to the abstract, especially in a technical field such as mathematics. One does not fully understand a concept or its larger significance until one has worked his way through a number of non-trivial examples in detail and gained familiarity with how things behave from first-hand experience, not from being told so (this is why it is necessary to solve many homework exercises by oneself). Needless to say, the historical development by which the concepts themselves were originally discovered follows a similar pattern. The historian of science Thomas Kuhn draws attention to these elementary facts about learning in his theory of scientific revolutions with his idea of a ‘paradigm’, in its proper and most appropriate sense of an exemplary solution of a problem (not the loose sense in which the word is often bandied about these days, to mean a whole constellation of ideas or ‘Weltanschauung’ as in Einstein’s paradigm of relativistic physics). In fact, before Kuhn raised the term into prominence, it had a rather limited meaning in grammar; a paradigm is a schematic declension of a verb, such as the high-school student of Latin will have encountered. Now, logically of course measure and integration could be taught by starting with the most general case and some economy would thereby be gained, as one finds in, say, Folland and Rudin (‘papa Rudin’, that is). But from a pedagogical point of view the supposed advantage of such an approach must be largely illusory. Repetition is of the essence of learning, after all. This reviewer finds it helpful to see concepts such as outer measure introduced first on the real line, where one can visualize everything. Moreover, chapter five on differentiation and integration is specific to the real line; the fundamental theorem of calculus in the form it receives from the point of view of Lebesgue theory makes sense only in one variable anyway.
Let us return to reviewing Royden after this excursus. Chapter six on the classical Banach spaces (i.e., Lp spaces in the unit interval) is a good illustration of the material of real analysis when it goes beyond what the student will have seen in introductory calculus and analysis. Royden provides easy-to-follow proofs of the Minkowski inequality, Hölder inequality, Riesz-Fischer theorem (establishing completeness) and Riesz representation theorem for bounded linear functionals. At this point, the student should be ready to reproduce Royden’s argument in the case of the spaces of convergent sequences and of sequences convergent to zero.
Before taking up general measure theory in Part III, Royden launches into an exposition of abstract spaces in Part II. In part, this entails a review of familiar concepts on metric spaces and point-set topology for anyone who has gone through baby Rudin, but here they are pushed farther in sections on Baire category, the separation axioms, compact and locally compact spaces, the Stone-Čech compactification, the Stone-Weierstrass theorem for algebras of continuous functions in compact Hausdorff spaces, the closed graph and Hahn-Banach theorems, weak topologies and reflexivity and a little on topological vector spaces (through a somewhat abbreviated proof of the Krein-Milman theorem apparently taken from Kelley’s book on general topology). Some reviewers will object to Royden’s tendency to relegate what are substantial results in real analysis to the exercises, such as Urysohn’s lemma, Tietze’s extension theorem and the Urysohn metrization theorem. But a beginning graduate student should be able to handle them, and hints are provided anyway. The sections on topological vector spaces and Hilbert spaces are too rushed; they barely scratch the surface of what could be extensive fields in mathematics. There would be no space in a textbook on real analysis to cover such topics adequately, however, so perhaps one should be content with merely gaining some initial exposure to why these ideas matter for the sake of that elusive quality, mathematical maturity.
Given the extensive preparation the student will have received in Parts I and II, the presentation of abstract measure and integration in Part III can proceed at a fast clip. To alert the reader to what new phenomena can arise, compared with the case of standard Lebesgue measure on the real line, there are sections on signed measures, the Hahn decomposition theorem, the Radon-Nikodym theorem and the Lebesgue decomposition theorem. The standard results in general Lp spaces are recovered in the space of four pages. After this quick march through measure and integration theory in chapter eleven, chapter twelve considers methods of defining a general measure on a sigma-algebra, via outer measure and the Caratheodory extension theorem. Here, we also get sections on product measures and the Fubini and Tonelli theorems. The concepts are illustrated by means of a nice section on integral operators from Lq to Lp. The chapter is rounded out with sections on inner measure, Caratheodory outer measure and Hausdorff measure. The attentive student should take notice of what is being done here; what are standard results that one takes for granted in multivariable calculus, such as changing the order of integration, have to be established in the setting of general measure spaces.
This reviewer rather enjoyed the following chapter thirteen, in which Royden enters into some of the non-trivial complications that arise in locally compact Hausdorff spaces, with the concepts of Baire measure and regularity through the Riesz-Markov theorem for positive linear functionals on the space of continuous functions with compact support and the Riesz representation theorem for bounded linear functionals on the space of continuous real-valued functions on a compact Hausdorff space. The coverage in this chapter is extensive enough to get to some substantial results; the interested reader can find more in Halmos’ dated but classic text on measure theory.
In chapter fourteen, Royden applies the techniques developed so far to prove the existence of a Haar invariant measure on locally compact topological groups, making essential use of the Hahn-Banach theorem. The material on topological groups, group actions, unicity of the invariant measure and groups of diffeomorphisms comes across as too compressed to be really satisfactory, but it does illustrate some advanced applications of real analysis.
This reviewer fails to appreciate what chapter fifteen on mappings of measure spaces, Boolean sigma-algebras, measure algebras and Borel equivalence is all about. Suffice it to say that we get some further non-trivial applications of concepts in real analysis, through the Kuratowski theorem on Borel equivalence and a characterization of isometries on Lp[0,1]. The reviewer does appreciate the concluding chapter on the Daniell integral, though. The Daniell integral is a way of extending the usual Riemann integral to a larger class of functions without going through all the apparatus of measure and Lebesgue theory with the aid of a suitable positive linear functional which, nevertheless, enjoys all the desirable properties of the Lebesgue integral with respect to convergence of sequences of functions. The Daniell integral is probably not a canonical topic in real analysis and one could very well get along without it; still, it is nice to see it disclosed as a possibility.
Royden’s text is liberally supplied with homework exercises, some 620 in all over fourteen chapters. As a rule, they are either not too difficult or, in the case of results that might be more of a challenge to the beginning student, outfitted with a hint or plan of proof to follow. They do, however, illustrate the material nicely and it is a good idea to attempt many or most of them in order to solidify one’s acquaintance with real analysis.
If the reader is patient and sticks with it, he will end up with an exposure to real analysis at a somewhat advanced level, though not as far advanced as that of papa Rudin, say. The level attained, though, suffices to go on to functional analysis proper and the world of interesting applications it opens up (in mathematical physics, the theory of partial differential equations, probability theory, operator algebras etc.). The reviewer will hazard a surmise which, technically, he is not entitled to make, not having studied Folland’s competing text thoroughly: the apparently greater sophistication with which Folland treats the subject of real analysis is, for the most part, window-dressing and not really necessary to an adequate command of the material, which is to say, adequate for the important applications as opposed to a hypertrophic development merely for its own sake (one encounters a similar phenomenon in set theory, which most mathematicians need as basic material to use in their own disciplines but which a small band of devotees has extrapolated into a baroque monstrosity largely devoid of significant applications to real, everyday mathematics outside its own narrow domain). The reviewer does hope to work his way at least through the first half of Folland eventually, when time permits, so keep posted!
January 29, 2013
If I can, I would give it one and a half. I read only seven chapters of the book.The merits are that it is a slow introduction to Lebesgue measure and integration. On the other hand, a lot of non-trivial theorems are left to the reader, and the author proves only very simple theorems. It feels like a math problem book, without solutions. Folland's book is much better than this one.
August 6, 2018
Nice book for me, I like this
This entire review has been hidden because of spoilers.
May 14, 2025
QED. Fk measure theory. I think I’m mostly ashamed because I ended up finding it sort of interesting once I got glimpses of how it relates to probability theory.
Favorite exercise: showing that continuous functions map F-sigma sets to F-sigma sets. (Starting with an open cover of the function mapping itself, projecting down onto the domain, using Heine Borel to find a finite subcover THEN mapping back up to the function was so counterintuitive but v clever).
Least favorite part: Why tf we using Chi for characteristic functions. Just use a fancy 1 like a normal person.
Overall, proofs could’ve been a little more ethical. But it was not a bad textbook all in all. Draw a damn diagram from time to time though.
Favorite exercise: showing that continuous functions map F-sigma sets to F-sigma sets. (Starting with an open cover of the function mapping itself, projecting down onto the domain, using Heine Borel to find a finite subcover THEN mapping back up to the function was so counterintuitive but v clever).
Least favorite part: Why tf we using Chi for characteristic functions. Just use a fancy 1 like a normal person.
Overall, proofs could’ve been a little more ethical. But it was not a bad textbook all in all. Draw a damn diagram from time to time though.
May 14, 2017
Only read the chapters on measure theory, integration and introduction to classical Banach spaces, according to school syllabus. The writing in 2nd/3rd edition seems better than the 4th edition for some reason, possibly due to the flow. As I am not working in this field, probably won't go beyond these topics anymore in near future.
Otherwise, the book is extremely clear in introducing measure theory and function spaces. It is probably one of the few "standard" useful texts in analysis.
Otherwise, the book is extremely clear in introducing measure theory and function spaces. It is probably one of the few "standard" useful texts in analysis.
Read
February 20, 2016A classical in Real Analysis
September 10, 2025
only 4 chapters in thus far but my god is this textbook lacking in motivation, examples, and-most shockingly- proofs! Leaving the open set definition of measurability as an exercise for the reader is genuinely sacrilege.
February 16, 2018
great book
July 27, 2021
Definition heavy, example light. I would flip the proportions. Also light in it's treatment of probability distributions, which is my favourite application of measure theory.
December 27, 2022
Just approximate it with simple functions this shit is easy
May 27, 2025
Idk why hilbert spaces are barely covered, but the measure theoretic stuff made me wet
January 5, 2022
I have read this book for at least three time. This is the one of the best book for rookies to learn real analysis and help ones establish solid foundation for probability theory.
May 11, 2023
This text is riddled with typos, and this is true even AFTER Fitzpatrick went over it and fixed a bunch of them. It also supplies very little in the way of explaining why it presents the material that it presents -- which is quite standard for graduate texts, but leaves me a little unsatisfied.
It's a decent reference text, but I think Axler's text Measure, Integration, and Real Analysis is better.
It's a decent reference text, but I think Axler's text Measure, Integration, and Real Analysis is better.
July 15, 2007
This, together with Rudin's "Real Analysis", is one of the standard texts on the subject. I personally like Royden a little more -- it has a slightly more conversational tone (but not overly so), and it covers more functional analysis than Rudin does.
October 4, 2008
Didn't care for the way this book was organized. The first half was a bunch of theorems on the real line, and the last half was all almost identical theorems on arbitrary topological spaces. I've started Rudin's and so far feel like I'm getting more out of that.
October 25, 2007
good measure theory book, title is misleading, but this is a classic text. a great book to learn from
June 27, 2008
I learned how to construct the Reals from the Integers.
Want to read
April 13, 2010possible text for math 525a
May 11, 2010
It's the standard for graduate analysis texts but not the best for learning on your own. Some major topics are covered only in the exercises. Worst. Index. Ever.
August 30, 2011
One of the must-haves for a beginner in learning analysis; concise and not deep.
January 22, 2014
this proof is trivial...is the main text of the book. was left confused and baffled a bit more than most math books
Want to read
May 4, 2023Real Analysis - bit more deep
Read
May 17, 2010I only read chapters 2-6.
Displaying 1 - 23 of 23 reviews