Looking for a rigorous treatment of improper multiple Riemann integrals

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.

How to prove that $\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$?

$1989 \mid n^{n^{n^{n}}} - n^{n^{n}}$ for integer $n \ge 3$

How i show this beautiful inequality :$\frac{x^n}{x^m+y^m}+\frac{y^n}{y^m+z^m}+\frac{z^n}{z^m+x^m}\geq \frac{3} {2}(\frac{1}{\sqrt{3}})^{n-m}$?

Will the Lebesgue integral of a real valued function always be a Riemann sum?

Is a symmetric positive definite matrix always diagonally dominant?

Vector Space Structures over ($\mathbb{R}$,+)

Find the maximun of the sum $\sum_{k=1}^{n}(f(f(k))-f(k))$

Recreational problems in set theory?

Closed form for the infinite product $\prod\limits_{k=0}^{\infty} \left( 1-x^{2^k} \right)$

Examples of classes $\mathcal{C}$ of structures such that every finite group is isomorphic to the automorphism group of a structure in $\mathcal{C}$

Proving a curious formula of $\pi$?

if $\left | x_{n+1}-\frac{x_{n}^2}{x_{n-1}} \right |\leq 1$, show that $(\frac{x_{n+1}}{x_{n}}) $ convergent