Is there a Borel function $f:2^\omega\to\omega^\omega$ such that for every nonempty closed perfect set $P\subseteq 2^\omega$, $f|P$ is a dominating family of functions in $\omega^\omega$?


This is a refinement of the question I asked in: [A Borel perfectly everywhere surjective function on the Cantor set](https://mathoverflow.net/questions/396368/a-borel-perfectly-everywhere-surjective-function-on-the-cantor-set). I think the argument which yields a "no" answer there would also give that there is no function $2^\omega\to\omega^\omega$ with comeager, or possibly even nonmeager, image on every perfect set. But while every nonmeager family of functions in $\omega^\omega$ is unbounded, the converse (even for dominating families) is not true.



A Borel perfectly everywhere dominating family of functions
The answer is no. By Lusin's Theorem there exists $A \subseteq 2^\omega$ closed and of positive measure such that $f \restriction A$ is continuous. Since $A$ cannot be countable it must contain a perfect set $P$. Now $f[P] \subseteq \omega^\omega$ is compact, hence bounded in $\omega^\omega$.


