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I am a master's statistics student who is just about to finish studying Terence Tao's Analysis I and Analysis II books and I really liked his way of teaching things. Besides its content, I also need to study measure theory for researching purposes. Based on my needs and background, I would like to know if it is a good idea to study Terence Tao's An Introduction to Measure Theory or should I reinforce my theoretical basis before doing so.

If this is the case, I am very keen to study the topics covered by Rudin's Principles of Mathematical Analysis in order to complement what I have learned so far, but I would like to know first if there is a similar book which covers the same topics and is written in the same or similar way as Tao's does.

I am also interested in any book recommendation related to measure theory, but once again I'd prefer texts written similarly to the way Terence Tao's does.

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  • $\begingroup$ I actually just answered a similar question here $\endgroup$ Commented May 31, 2020 at 22:37
  • $\begingroup$ Thanks for the references! Could you also provide an analysis book other than Rudin's Principles of Mathematical Analysis covering the same topics? $\endgroup$
    – user0102
    Commented May 31, 2020 at 22:44
  • $\begingroup$ Abbott's "understanding analysis" is apparently quite good. I think after Terry Tao's analysis books, though, you should be well prepared to read Rudin. $\endgroup$ Commented May 31, 2020 at 22:46
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    $\begingroup$ After Analysis II you already know some measure theory / Lebesgue integration and you're ready to dive into An Introduction to Measure Theory. Just dive in. You don't have to read baby Rudin first. You can always backtrack and fill in gaps if you need to. $\endgroup$
    – littleO
    Commented May 31, 2020 at 23:02
  • $\begingroup$ Thanks for the contribution @littleO. I will proceed as you have suggested. $\endgroup$
    – user0102
    Commented Jun 1, 2020 at 1:33

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The answer to your question is YES: Terence Tao's Analysis I and II provide you with all of the necessary knowledge to complete a graduate measure theory course. You do not need to go over other books.

Having said that, it could help you a lot to have a good intuition in topology when dealing with Lebesgue measures. Also, there are some extra difficult exercises, e.g. Exercise 1.1.11 would require some knowledge of abstract linear algebra and linear isomorphisms, or in Exercise 1.1.17 knowledge of polyhedral geometry is kind of necessary; but you can complete the course without having to solve these.

Finally, if you are still having difficulties, you can consult solution manuals here.

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