Fix a probability distribution (measure) $\mu$ and define $$ T_f(\mu) = \{\nu : f_*\nu=\mu\}, \quad T_\nu(\mu) = \{f : f_*\nu=\mu\}. $$ Here, $f$ is function between measure spaces, $\nu$ is another probability distribution, and $f_*\nu$ is the pushforward of $\nu$ under $f$. $T_f(\mu)$ (resp. $T_\nu(\mu)$) can be interpreted as the inverse image of the map $\nu\mapsto f_*\nu$ (resp. $f\mapsto f_*\nu$). The set $T_\nu(\mu)$ in particular arises in optimal transport (OT), as the domain of the Monge problem.
I am interested in characterizations of these sets: When are they nonempty, unique (i.e. a singleton), infinite, etc. Since they play such a prominent role in OT, I expected to find some discussion in the classical references on OT but did not find anything. I would love a pointer to a paper, book, or notes that discusses any of these questions (or even other structural characterizations of these sets).