Context
I am having difficulty finding the posterior distribution of a Bayesian model with two parameters, which involves evaluating a double integral over an unbounded region. I prefer not to post the model for now, since this question is not strictly related to this problem in particular, but I probably will end up looking for help in another question.
Question
I have been taught that Fubini's Theorem holds if the region of integration is bounded, but all the books in Statistical Inference I own and my teachers always evaluate such multiple improper integrals (expectations, densities, and so forth) as iterated single improper integrals, and most of the times those are Gamma, Beta or density functions.
I recall from multivariable calculus that if a function $f(x,y)$ is positive and $D = [a,\infty)\times[b,\infty)$ then, if the limit exists:
$\int_D f = \lim_{(c,d) \to (\infty,\infty)}\int_b^d\int_a^c f(x,y) \, dx dy = \lim_{(c,d) \to (\infty,\infty)}\int_a^c\int_b^d f(x,y) \, dy dx $
Which I think it is not quite "the product" of two single improper integrals, so there must be something I am missing. Whenever I have applied this theorem in the context of multivariable calculus (homework, exams) $f$ was always a carefully chosen function with anti-derivatives. However, as I have mentioned before, the improper double integrals I have come across in Statistical Inference are not that easy.
So, for instance, would it be correct to do something like this?
Let $f(x,y) = g(x)h(y)$, $D = [0,\infty)\times[b,\infty)$ and $g(x) = x^{\alpha-1}e^{-x}$:
$\int_D f = \int_b^\infty\int_0^\infty f(x,y) \, dx dy = \Gamma(\alpha) \int_b^\infty h(y) \, dy $
If so, how can it be justified?
Sorry for my imprecision and thanks in advance!