Let $(X, \mathcal X)$ be a measurable space and $f:X \to \mathbb R \cup \{\pm \infty\}$ Borel measurable. Let $f^+ := f \vee 0$ and $f^- := (-f) \vee 0$. There exist two non-decreasing sequences $(f_n^+)$ and $(f_n^-)$ of non-negative simple functions such that $f_n^+ \uparrow f^+$ and $f_n^- \uparrow f^-$ pointwise. Let $f_n := f_n^+ -f_n^-$. Then $f_n \to f$ pointwise. We have $$ \begin{align} (f_n)^2 &= (f^+_n)^2 - 2f_n^+ f_n^- + (f^-_n)^2 \\ &= (f^+_n)^2 + (f^-_n)^2 \\ &\le (f^+)^2 + (f^-)^2 \\ &= (f^+)^2 - 2f^+ f^- + (f^-)^2 \\ &= (f^+-f^-)^2 = f^2. \end{align} $$
It follows that $|f_n| \le |f|$ and thus $|f_n| \uparrow |f|$ pointwise.
Could you confirm if my understanding is correct?