Measurability of a function taking countably infinite many values

Question

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!

Answer 1

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.