For $1 \leq i \leq n$ let $(\psi_{ij})_{1 \leq j \leq n_i}$ be sequences of random variables. Is there a better notation than
$$\{\psi_{ij} : 1 \leq i \leq n, 1 \leq j \leq n_i\}$$
to build a set from these random variables making the fact explicit that some of them might be a.s. equal?
The wikipedia article on the set-builder notation does not say anything about it. On this site I only found a notational solution for the inverse problem of marking that the elements used to build a set are distinct.
Doesn't the motivation for the $\{\ldots\}_\ne$ notation come from the fact that $\{\ldots\}$ does not prevent equality of variables listed inside the brackets? If you want to make sure your readers follow that interpretation, perhaps before the point where you start writing such sets you could explain that if you would ever mean $\{\ldots\}_\ne$ in the writing that follows, you would write $\{\ldots\}_\ne$ and not $\{\ldots\}$.