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I have the book Measures, Integrals and Martingales by Schilling for self studying. I find it good because the chapters are short and precise.

I need a recommendation for a book that will be optimal choice for reading before Schilling. The book should cover some topics on rigorous analysis. The book should me as short and precise as Schilings (I have bad experience with long books). So, for those of you who are familiar with Schilling: What is a good book to read in order to be well prepared to read Schillings book?

I have a Bachelor in Mathematical Economics from a Business School so I know a lot about linear algebra and calculus but from an applied point of view ("understand the formulas and compute with parameters" approach). Now I try learn mathematical analysis from a theoretical point of view and a are interested in books directed for pure math students

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    $\begingroup$ Maybe a standard introduction to Real Analysis textbook such as Understanding Analysis by Abbot, or one of the many similar books. $\endgroup$ – littleO Oct 8 '17 at 11:41
  • $\begingroup$ How about Calculus, 4th edition, by Michael Spivak ? I taught a course based on that book. $\endgroup$ – Glougloubarbaki Oct 8 '17 at 11:46
  • $\begingroup$ @littleO I have looked at the first chapters of Understanding Analysis! That is indeed a good book. I will now read that before continuing to Schilling, Thanks! $\endgroup$ – k.dkhk Oct 10 '17 at 18:17
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Royden, real analysis. It's not short but it goes direct to the point, it's quite rigourous and definitely simple as a first reading. Moreover it would be very good for you since you don't have any topological background, and some basic facts are well presented in the text.

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  • $\begingroup$ My library has it so I would definitely have a look. By looking at the Content table, it seems that some topics of Schilling (which I referred to in my post) is covered in Boyden. Do you see Boyden as a replacement for Schilling or as a starting before moving on to Schilling? Thanks for your suggestion! $\endgroup$ – k.dkhk Oct 8 '17 at 15:03
  • $\begingroup$ My experience is Cohn: springer.com/gp/book/9781461469551 is a very nice book on measure theory and a much better choice for self-study than Schilling. $\endgroup$ – g.s Oct 11 '17 at 1:39
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This was already mentioned, but without a doubt, I would highly recommend Royden and Fitzpatrick's Real Analysis, 4th Edition for measure theory. It not only covers Lebesgue theory, but also general measure spaces. It introduces some basic concepts in the first chapter as well. You could also try Walter Rudin's Principles of Mathematical Analysis (Baby Rudin). It's a classic text which covers the Lebesgue theory in its last chapter. However, it is incredibly terse, so it may not be the best for beginners.

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  • $\begingroup$ The 4th edition introduced a huge amount of errors in the text; I would use any other edition. $\endgroup$ – Michael Greinecker Oct 11 '17 at 4:34

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