How does $|fg|$ compare to $|f||g|$ for two measurable functions $f$ and $g$ on $\mathbb{R}^d$?

Question

Suppose I have two measurable functions $f$ and $g$ on $\mathbb{R}^d$.

I believe that we have $|fg| \leq |f||g|$. However, if $f$ and $g$ are both non-negative, then $|fg| = |f||g|$.

Is this correct ?

Thanks!

Answer 1

$|ab|=|a||b|$ for all real numbers $a,b$

Therefore

$|ab|=|a||b|$ for all measurable functions $a,b$ on $\mathbb R^d$