Measure, Integration & Real Analysis

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1. painintheazzApr 2020
   Like the others said you want to know basic linear algebra and probability. Another thing very helpful in general is mathematical analysis. From elementary real analysis up through measure theory and Lebesgue integral, at least. From here a lot of advanced math will open up to you: for example, big-boi probability theory and statistics.

   Here below is an intro to how math people think (FREE):

   * Book of Proof by Hammack: https://www.people.vcu.edu/~rhammack/BookOfProof/

   For intro linal and probability, try

   * Introduction to Probability by Bertsekas, Tsitsiklis: https://www.amazon.com/Introduction-Probability-2nd-Dimitri-...

   * Linear Algebra by Kuldeep Singh: https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh-ebo...

   Both (clear and easy to read) books above are on l#ibgen, I believe. Also, there's a free online course based on Bertsekas'/Tsitsiklis' book on edx if I recall correctly.

   I prefer textbooks to online stuff when it comes to math (especially analysis). The many elementary analysis books that I know (and like) are print-only.

   Although, you could probably do the following one after the other:

   For pre-analysis, try

   * From Calculus to Analysis by Schinazi: https://www.google.com/books/edition/From_Calculus_to_Analys...

   For modern analysis, try

   * Measure, Integration and Real Analysis by Axler: https://link.springer.com/content/pdf/10.1007%2F978-3-030-33...

   The one by Schinazi is on l#ibgen and the one by Axler is available free.

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2. VovaPutinDec 2019
   > math doesn't really go stale

   I apologize for taking this out of context, but I'd thought I'd comment. Even though math in textbooks doesn't change, presentation totally does.

   For example, "Measure, Integration & Real Analysis" by Axler [0] is a beauty to behold. Honestly, no books from as recent as aughts match the pics, the font, the color of this book...even the explanations and examples leave many (if not most) real analysis textbooks of the past and present in the dust.

   [0]

   The Book Page: http://measure.axler.net/

   Direct Link: https://link.springer.com/content/pdf/10.1007%2F978-3-030-33...

   It's free.

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3. throwawaymathDec 2019
   I have to imagine you know this if you're reading a book like that; but for anyone else reading this comment: I recommend you don't read Measure, Integration and Real Analysis for a first introduction to real analysis. It's "fun" to see Axler present measure theory first and foremost, but analysis is typically a prerequisite to that.

   The more pedagogically appropriate way to learn analysis is with heavy emphasis on basic topology (metrics) and with the language of geometry. Geometry is useful in calculus and analysis for intuition. Measure is beautiful and helpfully simplifies/generalized a lot of results, but it's really abstract and (in my opinion) unmotivated if you've never encountered rigorous integration or differentiation before.

   I'd recommend Tao's, Pugh's or Stein & Shakarchi's analysis books over this one if the focus is analysis. I agree with your broader point that the structure and presentation of math can substantively change, even if the fundamental content doesn't at the undergraduate and graduate levels.

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