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I am thinking of self studying the first six chapters of Folland's Real Analysis: Modern techniques and Their Applications. I had read the first six chapters of Baby Rudin in the first real analysis course I had taken and would love to hear what people think of Folland's book for a second real analysis course. Has anyone read this book as an undergraduate? Is it too challenging for an undergraduate student?

Bonus: Does anyone have any other suggestions for a different textbook that can be used in a second semester of real analysis? I've read about Spivak's book but I don't think a physics-based analysis course is relevant to me (I want to pursue graduate-level statistics in a couple years).

Edit: I would love to learn some measure theory.

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  • $\begingroup$ @peek-a-boo Is the book you've mentioned accessible to someone who's read the first six chapters of Baby Rudin? $\endgroup$ Commented Apr 30, 2020 at 5:14
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    $\begingroup$ I think it's a great book on measure theory that takes a fairly abstract approach. (Great in some ways, but not great at motivating things.) You're ready for it, and if you don't like it you can just switch to a different book. No need to commit to a particular textbook. You might want to complement it with a book that takes a more concrete approach to developing Lebesgue integration, such as Zygmund and Wheeden or Royden. Sheldon's Axler's new book on real analysis looks good. $\endgroup$
    – littleO
    Commented Apr 30, 2020 at 5:20
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    $\begingroup$ It's been a long time since I read Folland, but in my memory it is very good but a bit terse - occasionally lacking motivation and seeming a little too optimized for short proofs. I found Stein and Shakarchi to be a little more readable. But you can't go wrong with either, really. $\endgroup$ Commented Apr 30, 2020 at 5:29
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    $\begingroup$ My new book Measure, Integration & Real Analysis may suit your needs well. The electronic version of the book is available for free at link.springer.com/content/pdf/10.1007%2F978-3-030-33143-6.pdf . $\endgroup$ Commented May 1, 2020 at 4:42
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    $\begingroup$ @SheldonAxler Thanks for your suggestion! I glanced through your book and I'm quite tempted to read your book more than any other book on the subject. But, I am a little apprehensive about attempting the exercises in your book since I won't be able to tell if I've attempted a proof correctly. Neither the odd nor the even numbered exercise have solutions at the back of the book. Are you planning to release a(n) instructor's/ students' solutions manual (or some other resource of similar nature) for your book sometime in the near future? $\endgroup$ Commented Jun 6, 2020 at 21:50

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I read first few chapters of Folland's Real Analysis during summer after my senior year at Ohio State to prepare for a rigorous PhD program that I was soon to begin (with one focus being pure / applied econometrics). I had previously also read Baby Rudin, ch. 1-7 for a course (you'll need ch. 7 since uniform convergence of sequences of functions is important for measure theory). Folland is a bit terse, but precise, and it has been the primary text for first year grad analysis at many universities because its coverage/presentation is fairly traditional and complete: measure theory, basic functional analysis, topology that you need for analysis if you haven't taken a course in general topology), Fourier analysis, distributions. And one of the other reviewers is correct: you need to read at least Ch. 1 - 6 (to get Lp spaces), and Ch. 8 (Fourier Analysis) is very helpful for those who will take probability theory like you). Ch. 7 is a "bonus" since almost no text at this level proves the major theorems in locally compact Hausdorff spaces as thoroughly as Folland (and you'll see other measure theory texts refer you to this chapter for its coverage). I agree Folland is a bit dry, but adequacy of coverage makes up for it. A more lively presentation of similar topics is first 3 parts of Serge Lang's "Real and Functional Analysis" (and part 4 on differential calculus in Banach spaces is well done too). I just found about Axler's book, and I'm looking forward to skimming it soon.

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Has anyone read this book as an undergraduate? Is it too challenging for an undergraduate student?

I personally read Folland after Baby Rudin. If you think the style of Baby Rudin is good for you, then you should be able to read Folland.

However, I think you should go a bit further in Baby Rudin before learning measure theory. Chapter 7 and 8 are typically taught in first year analysis courses.

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  • $\begingroup$ Can you please elaborate on why you are suggesting that I read chapters 7 and 8 of Baby Rudin before reading measure theory? $\endgroup$ Commented Apr 30, 2020 at 6:22
  • $\begingroup$ @Ricky_Nelson Chapters 7 and 8 of Baby Rudin covers things like uniform convergence and Fourier series. I would say uniform convergence is a prerequisite to study things like convergence in measure in measure theory. Fourier series gives a basic example of an orthogonal basis of a Hilbert space, an object studied in functional analysis. $\endgroup$ Commented Apr 30, 2020 at 16:44
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It isn't too challenging for an undergraduate. I remember it being quite impressive. I did some advanced things as an undergraduate; I was at Berkeley and got honors in mathematics. I've forgotten a lot of things, but I do remember the book fondly, and for one thing, I learned about generalized Cantor sets in there. I recommend working out with it if you're interested in real analysis. It's a good reference to have around also.

I took real analysis my first year in grad school. I think we used Wheeden and Zygmund. They should have some material on measure theory.

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