How to write the realisation of a function whose domain is a function?

Question

Let $A$ and $B$ be arbitrary sets with arbitrary elements $a\in A$ and $b\in B$ and power sets $\mathcal{P}(A)$ and $\mathcal{P}(B)$. Further, let $f:A\to\mathcal{P}(B)$ be a function. I want to define a real-valued function $g:f(a)\to\mathbb{R}$ that assigns one real number to each element $b\in f(a)$.

My question is as follows: how do I denote the realisation of the function $g:f(a)\to\mathbb{R}$?

I have considered three (unsatisfactory) possibilities:

1. Writing this function as $g(f(a))=x$ for some $x\in\mathbb{R}$ is unsatisfactory, since $g(f(a))\subseteq\mathbb{R}$ and therefore $g(f(a))\notin\mathbb{R}$;
2. Writing this function as $g(a)=x$ for some $x\in\mathbb{R}$ is unsatisfactory, since $g(a)=g(f(a))\subseteq\mathbb{R}$ and therefore $g(a)\notin\mathbb{R}$;
3. Writing this function as $g(f(a),a)=x$ for some $x\in\mathbb{R}$ is unsatisfactory, since $g(f(a),a’)$ is undefined if $a\neq a’$.

Any help will be much appriacted.

Thank you all in advanced.

Answer 1

You need a family of functions indexed by the elements of $A$. For each $a \in A$ you have a function $g_a$ whose domain is the subset $f(a)$ of $B$ and codomain is the real numbers.

I would suggest writing this out in words, as above, rather than trying to find a compact formal description your reader would have trouble parsing.