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I currently own copies of these texts by Royden and Angus Taylor, and want some help in deciding which text to choose for self study of measure theory. I have come across favourable reviews of both texts. Royden's book is deemed a classic but Taylor's book has a juxtaposition of approaches to integration via linear functionals and via measure which seems like a plus.

For some background, I am comfortable with the first 8 Chapters of baby Rudin. The reason I can't do them consecutively is that it would take too much time. My goal is to be well prepared for graduate courses on Applied Differential equations, Partial Differential Equations and possibly Functional analysis.

If I have to choose just one text, which book might be the better bet? I am open to other suggestions as well. Thanks.

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  • $\begingroup$ If you have to pick one I say go with Royden. The Taylor book seems to be of unknown quality, with only four Amazon reviews, but Royden is a sure thing. The material in Taylor on the Daniell integral may not be that useful, whereas Royden discusses some functional analysis topics that will be useful. But, I'd also say you don't have to pick just one -- look at them both and see which one seems more clear to you for each particular topic. The time you spend reading the words is negligible compared to the time spent understanding proofs, so you may as well use both books together. $\endgroup$
    – littleO
    Jul 19, 2017 at 14:45
  • $\begingroup$ @littleO Thanks. My only apprehension about Royden's book is that i heard it delegates some important theorems to the exercises. Is this true? and does this affect the readability of the book? $\endgroup$
    – Hikaru
    Jul 19, 2017 at 15:01
  • $\begingroup$ I'm not sure. But by the way, my favorite book for this topic is Stein and Shakarchi. You might also take a look at that one. $\endgroup$
    – littleO
    Jul 19, 2017 at 15:10
  • $\begingroup$ I have both books too. Royden covers much more ground, but Taylor is much gentler. When I first started studying measure and integration theory, I quite liked Taylor for the lengthy explanations. Nowadays I prefer Royden. I imagine the right choice depends on one's background. Since you've got a good background in analysis, you might try focussing on Royden and consulting Taylor when Royden seems too quick. $\endgroup$
    – aduh
    Jul 20, 2017 at 2:17
  • $\begingroup$ @aduh Thanks for the advice. This seems to be a fine compromise. :) $\endgroup$
    – Hikaru
    Jul 20, 2017 at 10:21

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I'll dissent slightly from the discussion so far by suggesting that I advise against trying to read Royden-Fitzpatrick cover to cover. I found it to be very verbose when I was learning from it. The first three chapters of Royden will get you a long way toward an education in measure theory on the real line: definitely take a while to absorb it, even doing the preliminaries in Chapter 1 as a refresher. After that you might consider another text for the general theory of measurable spaces, measures, and Lebesgue integration. Royden does this in his Chapter 4 in the context of Lebesgue measure on the real line and only later goes back and does this in the general setting, but I don't think it's worth your time to see this done in a special case and then have it reviewed later. Plus, it's not exactly true that Lebesgue measure is the only interesting (or elementary) example of a measure so I don't really agree with the strategy of Royden-Fitzpatrick.

How did your first encounter with "baby Rudin" go? If it went well, then you might appreciate the first chapter of Rudin's Real and Complex Analysis, where he introduces measures and Lebesgue integration in full generality. If you feel intimidated by Rudin, there are other texts that do this. I used Avner Friedman's Modern Analysis and Richard Bass's Real Analysis for Graduate Students when I was first learning these things as an undergraduate. I just think reading Royden exclusively is narrowing your point of view unnecessarily and quite possibly adding a lot of work.

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  • $\begingroup$ That Royden covers much of the same material twice bothered me as well. I did think about Rudin's Real and Complex Analysis because i really enjoyed baby Rudin, especially the clarity of his exposition. But my apprehension is that I haven't had a course on Complex Analysis so I am not sure if that might affect me midway through the book. $\endgroup$
    – Hikaru
    Jul 20, 2017 at 18:07
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    $\begingroup$ No, it won't be a problem. It's basically two books, one on real analysis, and another, on complex analysis. He does include complex-valued functions from the start, but you worked with baby Rudin so that's nothing new. $\endgroup$
    – user81375
    Jul 20, 2017 at 20:58
  • $\begingroup$ Great..I'll see if i can work through Rudin. $\endgroup$
    – Hikaru
    Jul 20, 2017 at 21:53

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