I'm trying to think about this from the Riemannian integration perspective so let me know if Lebesgue integration or something else is better. An example where I seem to be running into problems is with integration over the domain $\mathbb{R}\setminus\mathbb{Z}$. If you want to integrate $f\colon\mathbb{R}\rightarrow\mathbb{R}\setminus\mathbb{Z}$ given by
$\begin{equation} f(x) = e^{-x} \end{equation}$
over $D = (0, 4)$, you should be able to get away with Riemannian integration (even though the interval is discontinuous) since the discontinuities are negligible. I guess my intuition for this is that $f(x)\Delta{x}$ will disappear in the limit $\Delta{x}\rightarrow0$ for these discontinuities since they are finite. In a Riemannian sum, as the number of partitions goes to infinity, the contributions (or lack thereof) of 4 pieces will make no difference. So in that sense this integral over a finite domain seems to make sense, or be something we can in a way "get away with". But what about the integral from $0$ to $\infty$? In that, we have an infinite number of missing partitions. What would the difference amount to? How do you compute or even define integration here?