Is there a version of Fubini's theorem for improper Riemann integrals? Here's an example of what such a version might look like.
If $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is bounded and non-negative over the set $E\subseteq\mathbb{R}^n$ and if the improper Riemann integral $I:=\int_E f\left(\mathbf{x}\right)\ \mathrm{d}\mathbf{x}$ exists and is finite, then the following iterated integral is well defined and evaluates recursively to $I$:
$$ \int_{-\infty}^\infty\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty f\left(x_1,x_2,\dots,x_n\right)\mathbb{1}_E\left(x_1,x_2,\dots,x_n\right)\ \mathrm{d}x_n\cdots\mathrm{d}x_2\mathrm{d}x_1 $$
So, for almost every $\left(x_1,\dots,x_{n-1}\right)\in\mathbb{R}^{n-1}$, the function
$$ x_n \mapsto f\left(x_1,\dots,x_{n-1},x_n\right)\mathbb{1}_E\left(x_1,\dots,x_{n-1},x_n\right) $$
is non-negative, bounded and Riemann integrable on every closed interval $\left[a,b\right]\subseteq\mathbb{R}$ ($a<b$) and the function $g:\mathbb{R}^{n-1}\rightarrow\mathbb{R}$
$$ g\left(x_1,\dots,x_{n-1}\right):=\lim_{r\rightarrow\infty}\int_{-r}^r f\left(x_1,\dots,x_{n-1},x_n\right)\mathbb{1}_E\left(x_1,\dots,x_{n-1},x_n\right)\ \mathrm{d}x_n $$
is again bounded and non-negative over $E'$ = the projection of $E$ on the first $n-1$ coordinates, and the improper integral $\int_{E'}f\left(\mathbf{x}'\right)\ \mathrm{d}\mathbf{x}'$ exists and equals $I$.