In most applications of multidimensional integration I've seen, e.g. n-dimensional Fourier transforms, and in almost all the introductory and rather heuristic texts such as Riley, Hobson and Bence, we can rewrite the n-dimensional integral as
$\int_{\mathbb{R}^n}^{} \text{d}^nk = \int_{-\infty}^{\infty} \text{d} k_1 \int_{-\infty}^{\infty} \text{d} k_2 ... $
an iterated integral, if I understand correctly, this follows from Fubini's theorem, which I saw formulated just using the conditions that the integrand $f : [a, b] \times [c, d] \rightarrow \mathbb{R}$ is continuous. So do we just take this same argumentation and extend the limits (a, b, c, d) of the domain to infinity?
Further, would it make sense to have, say, a Fourier transform if we can't compute it like an iterated integral? Are there common integrals over $\mathbb{R}^n$ where we cannot use this property? If couldn't do this, could we switch to spherical coordinates? If yes, how would we get the relationship between the volume elements?