Functions whose codomain's form depends from the element taken from the domain

Question

As in the title, I have a notational problem regarding functions whose codomain directly depends on the elements taken from the domain.

Here there is an example.

Let $X$ be a set with $x \in X$. Let $Y$ be another set and put a correspondence between $X$ and $Y$ such that for every $x \in X$ there is at least one $y \in Y$. Let $Y(x)$ denote the set of those elements of $Y$ that are in correspondence with $x$.

Now, imagine that we want to set a function $f$ that for every $x \in X$ gives an element in the set of probability distributions over $Y(x)$.
How do we write this function?

An option is of course $$ f : X \to \Delta ( Y(x) ), $$ but in my opinion this is not completely correct, because of the $x$ that pops up in the codomain that does depend from the element we choose from the domain, without making explicit this relation.

Another option is to write that there is a function $f$ such that $x \mapsto \Delta ( Y(x) )$.

What is the correct one?
Is there any other option?

Thanks in advance for your feedbacks.

Answer 1

If I were going to be talking about only one such function, I’d probably just say something like this:

Let $f:X\to\bigcup_{x\in X}\Delta\big(Y(x)\big)$ be such that $f(x)\in\Delta\big(Y(x)\big)$ for each $x\in $X$.

If I were going to be talking about several such functions, I’d define the set of them:

Let $$D=\bigcup_{x\in X}\Delta\big(Y(x)\big)\;,$$ and let $$\mathscr{F}=\left\{f\in{}^XD:f(x)\in\Delta\big(Y(x)\big)\text{ for each }x\in X\right\}\;.$$

(I use the notation ${}^AB$ for the set of functions from $A$ to $B$; you can substitute whatever other notation you prefer.)

Answer 2

Let $X$ denote a set, and $Y$ denote a function that accepts an element $x \in X$ and returns a set $Y(x).$ Then the set of all functions that accept an element $x \in X$ and return an element of $Y(x)$ is denoted $\prod_{x \in X}Y(x)$. So

$$f \in \prod_{x \in X}\Delta(Y(x))$$

is the notation you're looking for.