Suppose that $f,g,h,l\in L^1$ are positive and normalized (PDFs) functions. Then, how can I prove the following "triangle inequality"?
$|f(x)g(y) - h(x) l(y)|\leq |f(x)-h(x)| + |g(y)-l(y)|$?
If this is true, is the positivity condition necessary or it can be dropped?
Your inequality follows by the following calculation.
$$|ab-cd|=|ab-cb+cb-cd|\le|a-c||b|+|c||b-d|\le|a-c|+|b-d|,$$ where $a=f(x)$, $b=g(y)$, $c=h(x)$ and $d=l(y)$.
The proof only uses the triangle inequality for numbers and that $b$ and $c$ have absolute value smaller or equal to 1.