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As the title says, I'm interested in reading about measure theory and integration theory on my own time, and was hoping for book recommendations, prerequisites, and things like that.

As for my background, I've taken introductory analysis, where we covered the "typical" subjects up until some theorems about differentiable functions.

I'm taking intermediate real analysis now; we talked about metric spaces in general for a while, but lately we've focused solely on $\mathbb{R}^n$. Now we're working towards a proof of the implicit function theorem in $\mathbb{R}^n$.

I haven't learned any of the theory behind Riemann integration, unfortunately. Is it important to be familiar with this first?

Ducky
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    It is not necessary to have seen Riemann integration.It might even help to not have seen Riemann integration first. Although later, when you are taught Riemann integration, you have to be extra careful with the properties you lose. – Pp.. Feb 05 '15 at 14:56
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    One important piece of motivation to remember when beginning to study the Lebesgue integral is its construction allows you to integrate more functions than are Riemann integrable (a function is Riemann integrable iff its set of discontinuities has measure 0 -- i.e., is small). With the Lebesgue integral, you can integrate more functions than this (and the Lebesgue integral agrees with the Riemann integral on Riemann integrable functions). I don't think you have to know Riemann integration to understand the Lebesgue integral. – layman Feb 05 '15 at 15:01
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    @user46944 In that if and only if theorem one must not forget the condition of being bounded. – Pp.. Feb 05 '15 at 15:03
  • To be honest, the theory of Lebesgue integration is cleaner than the theory of Riemann integration, so I don't think you'll have much trouble. The only important idea that is shared between the two is cutting an interval into small pieces, and that much is covered in calculus. – Ian Feb 05 '15 at 15:05
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    Related: http://math.stackexchange.com/questions/1583387/multidimensional-riemann-integration-and-notion-of-volume-or-lebesgue-theory-and/1583427#1583427 – Aloizio Macedo Jan 01 '16 at 15:48

4 Answers4

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I recommend "Real Analysis, Modern Techniques and Their Applications" by "Gerald B. Folland". The book is excellent for self-studying.

Good Luck :)

Sepideh Bakhoda
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    I would also recommend this book, but I would also give the (soft) warning that the OP will probably not understand some parts (for example the first chapter). But most of this chapter is only needed to understand some comments & notes at the end of the chapters, so it is important to read on. Also, I recommend to use math.SE for asking questions if the OP gets stuck. Another possibility (probably a bit more difficult) would be to try Rudin's "Real & Complex Analysis". – PhoemueX Feb 05 '15 at 15:07
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I recommend this link:

https://www.youtube.com/watch?v=fuIeczxyKNA

There are other videos on youtube that are very resourceful. You will see them after viewing this link.

Obinoscopy
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I would suggest "The Elements of Integration and Lebesgue Measure" by Bartle, http://www.amazon.com/The-Elements-Integration-Lebesgue-Measure/dp/0471042226, for a first course in measure theory.

It is aimed at people with your level of knowledge, has plenty of examples and isnt too concise. There are also plenty of questions at the end of each chapter

Trajan
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The schaum series" is the source of attraction mathematics and physics books, which is the best for everyone.THE "REAL VARIABLES" is the only book on measure and integration, Fourier analysis in this series.

Sima Midda
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