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I have never encountered measure theory or manifolds yet, despite being close to my third year university level. Any texts for either or both of these subjects would be greatly appreciated.

Analysis
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    Measure theory books: http://math.stackexchange.com/questions/308387/introductory-measure-theory-textbook?lq=1 ; http://math.stackexchange.com/questions/46213/reference-book-on-measure-theory – littleO Sep 05 '14 at 11:04
  • Also, just a really soft question for the comments, when might one usually encounter either of these topics? – Analysis Sep 05 '14 at 11:04
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    Manifolds books: http://math.stackexchange.com/questions/46482/introductory-texts-on-manifolds ; http://math.stackexchange.com/questions/14475/good-introductory-book-on-calculus-on-manifolds – littleO Sep 05 '14 at 11:05
  • littleO gave you several links. I encourage you to search the forum: I'm sure there are more. If none of the responses you found there are good for you, edit the question to explain why and be much much more specific about your needs. – Willie Wong Sep 05 '14 at 11:20
  • @WillieWong Despite the fact that this is a SEO gold mine... – Analysis Sep 05 '14 at 11:21
  • Also, measure theory and manifolds being such disjoint subjects as they are, your question really should have been two separate questions. (The two subjects meet again in geometric measure theory, but that's a quite advanced topic requiring solid foundations in both.) – Willie Wong Sep 05 '14 at 11:22
  • @WillieWong I see, that was part of my curiosity, I was told the two interacted heavily. – Analysis Sep 05 '14 at 11:24
  • @WillieWong do you perhaps have any opinion on the really soft question I asked in the second comment? – Analysis Sep 05 '14 at 11:28
  • What do you mean by "when"? – Willie Wong Sep 05 '14 at 11:30
  • Sorry, that was very unclear: Number of years into university level Mathematics? Final year undergraduate, first year grad etc – Analysis Sep 05 '14 at 11:35
  • @WillieWong I forgot to tag you, my apologies. – Analysis Sep 05 '14 at 11:49
  • It depends on university, course of study, etc. In my experience they are often introduced as a third year (American) or second year (European) undergraduate class. Though I have seen earlier and alter. – Willie Wong Sep 05 '14 at 15:36
  • @Analysis yeah there is a connection between measure theory and manifolds. The most accessible answer is to look at Sard's theorem. But the other comments are correct that measure theory on manifolds requires first a good understanding of both measure theory and manifolds. I think in Michael Taylor's book Measure Theory and Integration, chapter 11 deals with some of this topic. Otherwise you should look at geometric measure theory or even some of the metric geometry books--A Course in Metric Geometry by Burago, Burago, and Ivanov comes to mind. – krishnab Aug 09 '17 at 21:46

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For an introduction to measure theory, I personally am very fond of Terry Tao's book (the link is to his blog, and is absolutely legal. Note there's a free online pdf version).

I must warn you though, that he has a particular style. Mainly, the he will give you the necessary definitions, and will provide some of the harder proofs, but the majority of the theory is presented as carefully structured exercises.

(I write this here because I did not find a reference to this book in the answers to previous questions.)

Aahz
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