How should I rewrite the beginning of this post?
It seems people aren’t commenting since it’s difficult to understand.
Any suggestions are welcomed.
Maybe break up the first paragraph, something like this:
Does there exist a function $f:[0,1]\to[0,1]$ such that
the graph of $f$ is dense in $[0,1]\times[0,1]$, and
the collection of all the pre-images of each of the subsets of the range of $f$ under $f$, such that the pre-images are non-measurable in the sense of Caratheodory, is a [non-perfect dense set] in the collection of all subsets of $[0,1]$, and
$f$ is non-uniform (i.e. without [complete spacial randomness]) in $[0,1]\times[0,1]$?
I had trouble parsing 2), I don't know whether what I have written conveys what you intended.