Let $f,g \in L^2(\Omega)$, where $\Omega$ is an open bounded subset of $R^n,\; n\ge 1$.
We consider the two following properties:
(1) $\frac{f}{g} {\bf 1}_{\{x\in \Omega;\; g(x)\ne 0\}} \in L^\infty(\Omega),$
(2) there exists $c>0$ s.t: $|f(x)|\le c|g(x)|,\;$ for a.e. $x\in \Omega$.
It is clear that: $(2)$ implies $ (1)$.
My question concerns the converse: i.e. Is it true that $(1)$ implies $ (2)$? In other words, under (1), do we have $g(x)=0 \Rightarrow f(x)=0$ for a.e. $x\in \Omega$?
No. Take $f(x)=c$ where $c>0$ and $g(x)=0$ for every $x \in \Omega$. The function in $(1)$ is essentially bounded as it is identically equal to $0$ (the indicator function is $0$ for every $x$) and $(2)$ does not hold.