If $\displaystyle f_{n} \to f$ in measure then prove that |f_n|->|f| in measure.
Nope it means $\displaystyle \mu(|f_n-f|>\epsilon) \rightarrow 0$.
Have you heard of the reverse triangle inequality? $\displaystyle ||x|-|y|| \leq |x-y|$. Use it to prove that $\displaystyle \{||f_n|-|f||> \epsilon\} \subset \{|f_n-f|> \epsilon\}$.