I am preparing a lecture about Fubini's theorem.
For me, in "real life" the most common application of Fubini's theorem is to "change the order of almost-everywhere quantifiers". I.e. if $P(x,y)$ is a certain (measurable) property which depends on a pair of points $x$ and $y$, and I know $P$ holds for a.e. $x$ and then for a.e $y$, then $P$ holds for a.e. $y$ and then for a.e. $x$. However, the instances I am aware of are in the context of proving some more advanced theorem.
I am looking for a nice and instructive example of an application of this principle, say on the unit square $\left[0,1\right]^2$. This requires the property $P$ to not be symmetric, for otherwise there is no need to apply Fubini.
Thanks :-)