TITLE: fraction of $L^\infty$ functions
QUESTION [0 upvotes]: Let $\Omega $ be a bounded domain of $R^n$, and let $a, b \in L^\infty(\Omega) $ such that $ : \frac{a}{b} 1_{\{x\in \Omega/ b(x)\ne 0\}} \in L^\infty(\Omega).  $
Does the following implication hold ?
for $a.e. \;  x\in \Omega,$ we have : 
$$ b(x)=0 \Rightarrow a(x)=0$$

REPLY [0 votes]: No, certainly not. Notice that having the characteristic function completely ignores where $b(x)=0$, so therefore contains no information about this set. Consider $a\equiv 1$ and $b\equiv 0$. Then the function you described is just the zero function, but the stated implication fails in a big way.
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