Q:

Is there a non-random variable with Borel set preimage? 

Is there a non-random variable function ξ : Ω → R, for which the following statement is true? 
$\forall x ∈ R$ $ξ^{-1}(x) = \{ω| ξ(ω) = x\} \in \mathfrak B(Ω) $ 
I know that this is true for functions which suit "random variable function" definition, but are there any functions which are not "random variable", but fit this rule? 

A:

Take $(\Omega,\mathcal{F})=([0,1],\mathcal{B}_{[0,1]})$ and $\xi(\omega)=\omega+2\cdot1_N(\omega)$, where $N\subset [0,1]$ is a non-measurable set.  Then for each $x\in \mathbb{R}$, $\xi^{-1}(\{x\})$ is either empty or a singleton so that $\xi^{-1}(\{x\})\in \mathcal{B}_{[0,1]}$. However, $\xi^{-1}([2,3])=N\notin \mathcal{B}_{[0,1]}$.

