How to go about finding all $\mathcal{F}-$measurable functions

Question

Let $\Omega$ be uncountable and $\mathcal{F}:=\{A \subseteq \Omega: A \operatorname{or} A^{c} \operatorname{countable}\}$

Find: All $f: \Omega \to \mathbb R$ that are $\mathcal{F}-$measurable.

With lack of a better idea to start I suppose I should look at two cases:

Let $B\in\mathcal{B}(\mathbb R)$ be countable.

Let $B\in\mathcal{B}(\mathbb R)$ be uncountable.

But I honestly do not know where to start as there are so many ways in which to define $f$ that I cannot see an adequate way to represent them all.

Any help, even a hint, is greatly appreciated.

Answer 1

Let it be that $f$ is measurable.

If $A=\left\{ x\in\mathbb{R}\mid f^{-1}\left(\left(-\infty,x\right)\right)\text{ is countable}\right\} $ and $B=\left\{ x\in\mathbb{R}\mid f^{-1}\left(\left(x,\infty\right)\right)\text{ is countable}\right\} $ then $A$ and $B$ cannot be empty (it would contradict that $\Omega$ is uncountable).

Set $A$ has the property: $a\in A\wedge c<a\implies c\in A$.

Set $B$ has the property: $b\in B\wedge b<c\implies c\in B$.

If $a\in A$ and $b\in B$ then evidently $a\leq b$.

Further we have $\sup A=\inf B$.

(The existence of some $s\in(\sup A,\inf B)$ leads to a contradiction).

If $x=\sup A=\inf B$ then the sets $f^{-1}\left(\left(-\infty,x\right)\right)$ and $f^{-1}\left(\left(x,\infty\right)\right)$ are both countable.

For this note that $f^{-1}\left(\left(-\infty,x\right)\right)=\bigcup_{n\in\mathbb N}f^{-1}(-\infty,x-\frac1n)$ and $f^{-1}\left(\left(x,\infty\right)\right)=\bigcup_{n\in\mathbb N}f^{-1}(x+\frac1n,\infty)$.

So characteristic for the measurable function $f$ is that some $x\in\mathbb R$ exists such that the complement of the fiber $f^{-1}(\{x\})$ is countable.

(It is straightforward that a function having this property is indeed measurable)

Answer 2

Just an idea : (I wanted to put it as a comment, but the text was to long). I'll edit later when I'll be more convinced.

$f$ measurable $\iff \forall B\in \mathcal B(\mathbb R), f^{-1}(B)$ countable or $f^{-1}(B)^c$ countable $\iff \forall B\in \mathcal B(\mathbb R),$ $f^{-1}(B)$ is countable or $f^{-1}(B^c)$ countable$\iff$ there is $S\subset \Omega $ countable and there is $c\in \mathbb R$ s.t. $f(x)=c$ for all $x\notin S$.

I'm not completely sure about the last $\implies $.