Q:

Measurability of a function taking countably infinite many values

Let $f$ be a function from a measure space $(\Omega,\mathcal{A},\mu)$ to a measurable space $(E,\mathcal{E})$ (where $E$ is at least countably infinite) taking a countably infinite number of values in $E$. Can I conclude it is measurable, and how can I prove it?
Thank you in advance!

A:

No, this is certainly not true in general.  For instance, if $\mathcal{A}=\{\emptyset,\Omega\}$ and $\mathcal{E}=\mathcal{P}(E)$ then no nonconstant function is measurable.

