As a beginner to the subject on the paper I am reading, I found terminology in a sentence that looks like this:
"Let $f$ be a probabilistic map from $X$ to $Y$, and let $m\in[0,1]$. For any fixed $y\in Y$ and for any $x$ sampled uniformly at random from $X$, assume that the probability of $f(x)=y$ is $m$."
My questions:
I never encountered such a map before, but I have learned that its true form is $f:X\rightarrow\{\text{probability measures on }2^Y\}$, which makes sense (here's a seemingly matching definition which helps confirm, tho I did not refer to it much: Probabilistic functions, and the category Prob). Have I understood it correctly?
Now, assuming no. 1 is correct, if I remove the "sampled uniformly at random" part from the above definition, then I have learned also that "probability of $f(x)=y$" means exactly $[f(x)](\{y\})$, or in general I understand it to be $Prob(f(x)\in B):=[f(x)](B)$ for any $B\subseteq Y$. Is this understanding correct?
Now comes the italic part. What does it mean exactly? I cannot make sense of it at all, probably (pun intended) because my knowledge in probability theory is saddening. At first, I guess it is something like the answer in Meaning of "uniformly sampling"?, but then what does $Prob(f(x)=y)$ mean in this case? I mean this mathematically, not intuitively, as in a notation/definition that I can operate on.
To elaborate more on no.3, I think there are two "random" things here: The choice of $x$ and the choice of $f(x)$ (although it does not mean your everyday function). The second is, I thought, answered in no. 1 and 2. For the uniform $x$ sampling, I am thinking of course of the usual probability measure $|A|/|X|$ for $A\in 2^X$. However, still what is the probability that we want, mathematically? Is it just their product, which is $\frac{[E(x)]({y})}{|X|}$ ?
Much appreciated!