Looking for a rigorous treatment of improper multiple Riemann integrals I'm studying undergraduate-level differential and integral calculus and have recently come across the topic of improper Riemann integrals. I'm familiar with the concept for single-variable functions, but I haven't been able to find a rigorous exposition of improper integrals for functions of several variables.
What I mean exactly is the 'natural' generalisation of the Riemann integral which arises from dropping one or both of the hypotheses of


*

*The function being defined on a set which is not bounded

*The function itself not being bounded.


Being familiar with the equivalent concepts for functions $\mathbb{R} \to \mathbb{R}$, I feel that it wouldn't be too hard for me to generalize the concept from that scenario (which I have found to be a lot better documented) to that of functions defined on $\mathbb{R^{n}}$, but I'm looking for a sort of standard text on it. I just can't seem to find it.
Apostol's Mathematical Analysis was a promising source but, again, it only deals with the single-variable case. Marsden & Hoffman's Elementary Classical Analysis does have a short section on it, which I find too shallow and aerial. Finally, Rudin's Principles of Mathematical Analysis  doesn't seem to cover what I want either, only mentioning improper integrals for the single-variable case.
I am baffled that none of these supposedly standard texts cover the topic I'm after with depth. I can only hope that somebody out here knows any better sources.
 A: Munkres Analysis on Manifolds covers this in Chapter 3.15. Having now twice taught from Munkres textbook, I will say that this is the section I find the hardest. I constantly find myself at the chalk board saying "oh, and now we'll need another $\epsilon$" and "oh, I forgot to put take an exhaustion by compact subsets five minutes ago." Integrals of bounded functions on bounded domains is really clean and students follow it well, and when we hit the improper integrals is where the confusion sets in.
Hubbard and Hubbard Vector calculus, linear algebra, and differential forms covers this in Chapter 4.11. (Chapter numbers vary from edition to edition; the chapter title is "Improper Integrals".) When I read this a few years ago I thought "This is so much clearer! I am definitely using this in place of Munkres next time!" But next time hasn't come yet, so I can't say if the students agree.
A: Ok, I would recommend specially $\textbf{ ELON LAGES LIMA, Real Analysis vol. 1 & 2}$. This book is thorough on the subject. I link here a version in portuguese that I could find but I know it has been translated into english. 
You could also try $\textbf{ JEAN DIEUDONNÉ, Foundaments of Analysis}$ for which I unfortunately couldn't find a link. Either way any library with a vast stock on mathematics will have it. He was also a member of Bourbaki so you may want to try to search under Bourbaki too.
But again, Lages Lima is immho the best book to follow for what you're asking about. Good luck!
