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.
Best Answer
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.