Fubini's theorem is usually stated as $$\int_{X\times Y}f d(\mu \otimes \nu) = \int_X \int_Y f(x, y) d\mu(x) d\nu(y).$$
So it is always defined in terms of integrations over the whole spaces. I was wondering how to apply this for subsets though.
1.) So my first question is how I would use it if I want to integrate over some subset?
2.) My second question is if we can only use it for rectangular subsets of the form $A\times B$, with $A\in \mathcal{A}(X)$ and $B\in \mathcal{A}(Y)$ or if we can also use it for integrating over general subsets $C\in \mathcal{A}(X)\otimes\mathcal{A}(Y)$? If the latter is also possible, how do we do it?
PS: There is a question related to this here Questions about Fubini's theorem however the answers seem to contradict each other?
Let $C \in \mathcal{A}(X) \otimes \mathcal{A}(Y)$ be a general measurable subset. Write $A = \{x \mid (x,y) \in C\}$ to denote the set of points in $C$ along the $x$-axis. For each $x \in A$, let $B(x) = \{ y \mid (x,y) \in C\}$ denote the "slices" along the $y$-axis for a fixed $x$. (Exercise: show that $A$ is $\mu$-measurable and $B(x)$ is $\nu$-measurable.)
The generalized version of Fubini's theorem is
$$ \int_{C}f(x,y) (\mu\otimes\nu)(\mathrm{d}x,\mathrm{d}y) = \int_{A}\left[\int_{B(x)}f(x,y)\nu(\mathrm{d}y)\right]\mu(\mathrm{d}x). $$