I have a multiset $S$, in which elements $s$ occur. Each element $s_i$ occurs $m_i$ times.
How can I write a set $F \in S$, which is the set of elements $s$ which occur at least $n$ times in $S$?
For example: $ S = \{a,a,b,b,b,c,c,c,d\}$, with n = 3, $F = \{b,c\}$
$F = \{s||s| \geq n \}$
but I don't think this is right (next to looking ugly with the double $|$)
Does
$F \subseteq S, |s| \geq n \;\forall\; s \in F$
work?
That notation of writing $|s|$ to indicated the number of times an element occurs in a multiset is something I've never seen before, and doesn't occur anywhere in the Wikipedia article on Multisets. What they use is a "multiplicity function", $m(s)$, or $m_S(s)$ if you want to show the dependence on the particular multiset.
And in fact, in that article they define the support of a multiset as
$$ Supp(A) = \{x \in U \mid m_A(x) \gt 0\},$$ which is very close to what you want to do. So I'd recommend writing your set as
$$ \{s \in S \mid m_S(s) \ge n\},$$
although it might be a good idea to explicitly call out "where $m_S()$ is the multiplicity function" - I don't think the notation and terminology for multisets is particularly well-known among mathematicians.