Counter example for Random variable

Question

I read the result that if $|X|$ is random variable then $X$ need not be random variable.

So, I am looking for counter example.

Thank you for your time.

Answer 1

Take any non-measurable subset $A\subseteq\mathbb{R}$ and define $$ X(\omega)=\mathbf{1}_A(\omega)-\mathbf{1}_{A^c}(\omega),\quad\omega\in\mathbb{R}, $$ where $\mathbf{1}_A$ is the indicator function for the set $A$. Then $|X|=1$ is a random variable but $X$ isn't a random variable (why?).