Suppose we have a finite $A=\{a_i,\,i=1,2\}$ and a finite set $B=\{b_1,b_2\}$ and define a probability distribution over the set $B$. Denote this by $\Pi(B)$.
Suppose also that we define a function $S:A\to\Pi(B)$.
In many instances, I have seen the notation $s\in S$. Can we do this? If yes, what is its meaning?
My understanding is that the function $S$ maps elements of $A$ into $\Pi(B)$ and we can write $s(a_i)\in S(A)$. Do we abuse notation by writing $s\in S$?
Can we be more specific about $\Pi(B)$, for example, is it a set? Can we explicitly expand it? Can we write $\Pi(B)=[0,1]$?
I'd appreciate any help or hint.
These questions arise from studying the Bayesian game in Game Theory literature.