Let $\Omega_n$ be the set of equivalence relations on $\{0,1,2,\dots,n-1\}$, each eq. rel. being equally likely. Let $X$ be the number of classes, and $Y$ the size of the largest class.

Note that when $n\ge 4$, $X$ and $Y$ are not deterministic functions of eachother. For instance, the equivalence relation $$01\mid 23$$ has $X=Y=2$, and $$01\mid 2\mid 3$$ has $X=3$, $Y=2$, so $Y$ does not determine $X$.

The number of sample points in $\Omega_n$ is the Bell number $B_n$, which usually is not a perfect power (1,1,2,5,15,52,203,877,$\dots$).

So $X$ and $Y$ form an example of jointly distributed random variables, the understanding of which does not seem to require the understanding of the power of any previously-introduced measure space...

Let $\Omega_n$ be the set of equivalence relations on $\{0,1,2,\dots,n-1\}$, each eq. rel. being equally likely. Let $X$ be the number of classes, and $Y$ the size of the largest class.

Note that when $n\ge 4$, $X$ and $Y$ are not deterministic functions of eachother. For instance, the equivalence relation $$01\mid 23$$ has $X=Y=2$, and $$01\mid 2\mid 3$$ has $X=3$, $Y=2$, so $Y$ does not determine $X$.

The number of sample points in $\Omega_n$ is the Bell number $B_n$, which usually is not a perfect power (1,1,2,5,15,52,203,877,$\dots$).

So $X$ and $Y$ form an example of jointly distributed random variables, the understanding of which does not seem to require the understanding of the power of any previously-introduced measure space...

Edit: Or consider sparse random graphs, let $X$ and $Y$ be some quantities associated with social networks such as cohesion or clustering coefficient.