a relation between $f$ ,$|f|$ in measurability

Question

Suppose $f: \Bbb R \rightarrow \Bbb R $ is function s.t. $|f|$ is measurable

Is the $f$ measurable? (True or False) .

if $|f|$ is measurable and $\alpha $ is arbitrary then $\{ x \in \Bbb R | |f(x)| > \alpha \} $ is measurable.

Answer 1

False. Take $D \subset \Bbb R$ non-measurable and define: $$f(x) = \begin{cases} 1, & \mbox{if $x \in D$} \\ -1, & \mbox{otherwise} \end{cases}$$

Answer 2

Note that if the range of $f$ is contained in $\{-1,1\}$, then $|f| \equiv 1$ is measurable. Is every function $\mathbb{R} \to \{-1,1\}$ measureable?