I read on wikipedia (the only free source I could find that talks about the boundaries of integration) that the Fubini-Tonelli theorem can be applied only on double integrals that have rectangular domain of integration. I was wondering if it could be applied to domains of integration that could be decomposed into smaller rectangles (and by extension every domain that could be approximated by rectangles, so using a logic similar to the Riemann integration almost every domain). In particular wanted to apply the theorem for a integral
$ \int_{0}^{2} \int_{0}^{\lfloor y \rfloor} f(x, y) \, dx \, dy $
where two rectangles would be formed, one $0 \leq y \leq 1 \quad \text{and} \quad1 \leq y \leq 2$