We have recently learned Fubinis Theorem for Lebesgue Integrals in lecture.
However, I do not see how it is all that useful. Since in our formulation we consider a function $f$ from $ X \times Y \to [-\infty, \infty]$ to be measurable on that space regarding the product measure of two measures $\mu$ and $\nu$. Furthermore, if $f$ is integrable regarding the product measure of $\mu$ and $\nu$, than we are allowed to integrate $f$ first in $Y$ and than $X$ or vice versa and end up with the same value for the integral as if we would integrate over $X \times Y$. However, I do not see how one can in practice prove straight forward, that $f$ is integrable on $X \times Y$.
Any tips on how to use the theorem and show the integrability condition would be very much appreciated. (We have not discussed the Fubini-Tonelli Theorem which is more useful in my eyes.)
A great question! Recall that a function $f$ is integrable if and only if $|f|$ is. But the integrability of $|f|$ can be verified from Tonelli's theorem (the prequel to Fubini).
I hope this answers your question.