Is there a notation for the distribution sampled from a set

Question

Suppose we choose uniformly from the set S, is there a well accepted notation for this distribution? I imagine something like $X \sim U(S)$ or $X \sim Uniform(S)$? I want to use this to write that I choose iid random variables from a finite dimensional vector space, and it would be very compact to write $$X_1, X_2, ..., X_n\stackrel{iid}{\sim} U(\mathbf{F}_q)^n$$

Answer 1

Unsolicited advice: I feel it is better/clearer writing style to write things out in plain words.

I think $U(S)$ or $\text{Uniform}(S)$ are well-understood for finite sets $S$, but I would still write "where $U(S)$ denotes the uniform distribution over the set $S$" nonetheless (after the first instance of the notation) to eliminate any ambiguity.

I would use the notation $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} U(S)$ to denote i.i.d. samples from the aforementioned uniform distribution (do not use $U(S)^n$).