Is there a generalized Fubini's theorem, which allows one to conclude that, for example, $$\iint_D \frac{dA}{x^2 - 2xy + 1} = \iint_{-1}^1 \frac{dx \, dy}{x^2 - 2xy + 1} = \iint_{-1}^1 \frac{dy \, dx}{x^2 - 2xy + 1},$$ where $D = \{(x, y) : -1 \leq x \leq 1, -1 \leq y \leq 1\}$, or course. Note that the integrand is not continuous at $(1, 1)$ and $(-1, -1)$, so we cannot apply the standard Fubini's theorem. However, the set of discontinuities is finite (so it has zero measure). Yet, I do not think $1/(x^2 - 2xy + 1)$ is integrable, so we cannot use the measure-theoretic Fubini's theorem either.
Can we apply this (1st page) formulation of Fubini's theorem?
Since $x^2-2xy+1\geq (x\pm1)^2\geq 0$ (the sign is $+$ or $-$ depending on the sign of $x$), you can apply Tonelli's theorem.