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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.

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  • $\begingroup$ No one? Really? $\endgroup$ – Jake Parsons Apr 30 '15 at 21:07

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