Books in Real analysis

[Giulio]

I think there is a gap in the suggested books... there are no real analysis books... it is true that once you learn analysis from Rudin's principles yuo can master a measure-theoretic book in probability that provides a self-contained introduction to real analysis... but i think it is worth to spend some months on the subject of real analysis per se.... also b/c it is a beautiful topic...

what do you think about, e.g.:

Royden - real analysis
Wheeden & Zygmund - Measure and integral - An introduction to real analysis
Folland - Real Analysis Modern techniques and their applications
Rudin - real & complex analysis

[mj]

I haven't read any of those. I learnt measure theory using Weir's book on Lebesgue Integration.

[GeekySerge]

...also b/c it is a beautiful topic...

What do you mean with "b/c"?

what do you think about, e.g.:

Royden - real analysis
Wheeden & Zygmund - Measure and integral - An introduction to real analysis
Folland - Real Analysis Modern techniques and their applications
Rudin - real & complex analysis

Now I'm studying Real & Functional Analysis using the Royden book and I think it's good! Anyway, I'm taking this course at KAIST university in South Korea (I'm an exchange student) and the Prof. use this book.
Instead in my university in Italy (Politecnico di Milano) the Prof. of Real Analysis suggest these books:
- W. Rudin, Real and Complex Analysis, McGraw-Hill
- H.L. Royden, Real Analysis, Macmillan
- H. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag
- H. Brezis, Analisi Funzionale, Liguori

I don't know about the 2nd and 3rt but the other ones are nice!

[GeekySerge]

I haven't read any of those. I learnt measure theory using Weir's book on Lebesgue Integration.

Hi Mark, I've seen that you've made a PhD in pure mathematics at the MIT.
Can you told me what kind of courses have you taken there?
I don't know, maybe you've taken the Real Analysis course ans an undergrad at Oxford, but what about harmonic analysis, etc.?

Thks

[mj]

well it was a long time ago...

it just comes down to what they run at a high level in a given year.

[GeekySerge]

well it was a long time ago...

it just comes down to what they run at a high level in a given year.

Don't worry!
I asked you because I'm thinking of applying for a PhD and I would like to assess how MIT has changed his program along the years.

Anyway, thanks for the answer.

[bigperm]

I used Royden in grad school. It's a good book (with a bad index). Your experience depends as much on the teacher as anything, so if you can take a class then just do that regardless of the book. Royden/Folland/Rudin are graduate-level books (not sure about the others), so don't try to self-study from them until you do an undergraduate class.

I'd guess that undergraduate-level real analysis (at the level of Baby-Rudin) is sufficient to read a graduate probability text.

The analysis presented in graduate texts is too abstract for most "real-life" applications. What you gain in studying graduate analysis is more "mathematical maturity." E.g. once you prove Stone-Wierstrauss on a locally-compact topological vector space then learning Martingales doesn't seem so hard. Also, since e.g. functional analysis is well-established you see many examples of complete theory, from abstract definitions to tight proofs to interesting examples. This helps guide you in developing your own theories, which are always messy at first.