We have $f_{\alpha} : \mathbb{R} \rightarrow [0,+\infty]$ defined by $f_{\alpha}(x) = \dfrac{ \sqrt{|x|} }{1-x} 1_{[0,\alpha]}(x)$.
I would like to prove that : $$ \forall X \in \mathcal{B}([0,+\infty]), f_{\alpha}^{-1}(X) \in \mathcal{B}(\mathbb{R})$$.
Do you know how to do it ? Thank you in advance.