Distribution Functions of Measures and Countable Sets

Solution 1:

If $S$ is any systems of disjoint non-degenerate intervals on real line than it is countable. In particular, this is true for the system $S=\{g^{-1}(x); x\in A\}$, which has the same cardinality as $A$.

This follows from the fact that every interval $I\in S$ contains a rational number so you can get an injective map $S\to\mathbb Q$ by mapping $I$ to some element of the (non-empty) set $I\cap\mathbb Q$. The set $\mathbb Q$ is countable.

In the above proof we have obtained an injection by choosing an element from each set $I\cap\mathbb Q$. In case this is interesting for you, I should mention that we can avoid using Axiom of choice. It suffices to notice that we can explicitly write down some well-ordering of $\mathbb Q$ and then simply choose the element of $A\cap\mathbb Q$ which is minimal with respect to this well-ordering.

By explicitly constructing a well-ordering of $\mathbb Q$ I mean that we are able write a formula in language of ZFC which describes a well-ordering of $\mathbb Q$. To see this it is sufficient to find any bijection between $\mathbb N$ and $\mathbb Q$ (without using AC). Well-ordering can be "transferred" using the bijection.

We can get bijection $\mathbb N\to\mathbb N\times\mathbb N$, e.g. Cantor's pairing function. It can be easily modified to a bijection $\mathbb N\to\mathbb Z\times\mathbb N$. If we want bijection which has $\mathbb Q$ as the codomain, we simply "omit" fractions that appear more than once. E.g. if $f:\mathbb N\to\mathbb Z\times\mathbb N$ then we can get $g:\mathbb Q\to\mathbb N$ by putting $g\left(\frac{p}q\right)=|\{f(k); k<n\}|$, where $n\in\mathbb N$ is the preimage of the pair $(p,q)$, i.e. $f(n)=(p,q)$. (We assume that $p\in\mathbb Z$ and $q\in\mathbb N$ are relatively prime.)

To see that we do not need AC to select a rational number from each non-degenerate interval, see also this question: Open Sets of $\mathbb{R}^1$ and axiom of choice

BTW after posting this answer I found the same proof here: Every collection of disjoint non-empty open subsets of $\mathbb{R}$ is countable?