Customer Reviews

Real and Functional Analysis (Graduate Texts in Mathematics) (v. 142)

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Up to my knowledge, this is the only book that constructs the Lebesgue integral for functions to a general Banach-space instead of the real numbers (thus saving us from the unnecessary and esthetically dissapointing construction through positive and negative functions).
I don't know how Lang does it, but eerytime you'll pick up one of his books, you'll marvel at the beauty of mathematics !

It drove me up the wall, in my first course on measure and integration, that integration was first done for positive functions, then for real functions by writing them as a difference of positive functions, then complex functions in terms of real and imaginary parts. Why couldn't you just integrate real-valued functions
intrinsically, without the silly decomposition into positive and negative parts?

After that course, I found Lang's book. What a blessing to see that you can just integrate in infinite-dimensional spaces right from the start. I can't understand why virtually all books on integration theory still succumb to the "positive functions first" approach.

I've read several analysis books and this is one of the better ones that I have read. It covers a variety of interesting and useful topics and the exposition is clear. It's presentation is a bit more abstract than some others starting with some functional-analytic concepts before doing integration in that framework. However, if you want to study stochastic analysis, getting in this frame of mind will definitely help your understanding of stochastic integration. For a truly thorough understaning of the subject, I recommend purchasing this book as well as the somewhat easier "Lebesgue Integration on Euclidean Space" by Frank Jones - the two together cost about the same as Royden, Rudin, or the terrible book by Aliprantis.