# 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.

• No one? Really? – user235270 Apr 30 '15 at 21:07
• I would recommend $\textbf{LAGES LIMA, real analysis vol. 1 & 2}$, $\textbf{Dieudonn\'e, notes of analysis}$ or $\textbf{SPIVAK, calculus}$. There your subject is thoroughly covered by different levels of difficulty. I think Lages Lima is a nice approach which is not too easy nor too hard. Good luck! – astro Jan 31 at 23:02
• @Axstroo Spivak's calculus does not deal with even double integrals (and improper multiple integrals, of course). I'm not able to identify other books you mentioned; if those really deal with the subject the question asked, please post it as an answer with complete bibliographic information or links to somewhere containing such information. – Orat Feb 1 at 4:28
• Improper double integrals are treated in section 9.2 (pages 326-337) of James J. Callahan, Advanced Calculus: A Geometric View (Springer 2010). Two special cases in $\mathbb{R}^n$ are briefly treated in section 3.2 (pages 51-52) of G. E. Shilov & B. L. Gurevich, Integral, Measure and Derivative (Dover 1977). A general treatment is given in section 11.6 (pages 150-160) of Vladimir A. Zorich, Mathematical Analysis II (Springer 2004), and presumably also in the second edition of that book. – Calum Gilhooley Feb 5 at 17:43

## 2 Answers

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.

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!

• Thank you for posting an answer. I'm sorry that I don't understand Portuguese, but it seems that it doesn't cover improper multiple integrals at all. I guess the correct title of Dieudonné's books is Foundations of Modern Analysis. It deals with single integrals in section 8.7 and integrals along a path in section 9.6, but I couldn't find anything about multiple integrals. – Orat Feb 1 at 15:56
• Lages Lima deals properly with multiple integrals in chapters 8 and 9. It is of course somewhat of a theorical approach but the subject is sufficiently covered. I advise you ask for an english version at any good library specialized in math. If you want to take a look over chapters 8 and 9 I figure you will at least get a glimpse of how much of the subject is actually covered in the book. Good luck :) – astro Feb 2 at 16:32
• Although I don't check the English version yet, your reply sounds like you don't understand the OP. The question is about improper integral: a generalization of the Rieman integral, which is defined like $\lim_{n\to\infty}\int_{D_n} f(\boldsymbol{x})\,d\boldsymbol{x}$ (or perhaps $\sup_{D\subset\Omega}\int_D f(\boldsymbol{x})\,d\boldsymbol{x}$) for suitable sets $D_n$ or $D$. It is a very theoretical nature, and the problem is it's hard to find a (standard) textbook that deals with it in full generality. – Orat Feb 3 at 9:42
• Lol you're totally right xD I was so focused on answering a bountied questions that totally went off the track. Sorry! – astro Feb 7 at 19:00