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{\bf Question}

Show that if $f:{\bf R^n}\rightarrow {\bf R}$ is measurable, so is
$\ds \frac{1}{f}$ (where $\ds\frac{1}{0}=+\infty$).


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{\bf Answer}

$\ds \{x|\frac{1}{f}(x)<c\}=\left\{\begin{array}{lcc}
\{x|\frac{1}{c}<f(x)<0\}&{\rm if}& c<0\\ \{x|-\infty<f(x)<0\}&{\rm
if}&c=0\\ \{x|-\infty\leq f(x)<0\}\cup\{x|f(x)>\frac{1}{c}\}& {\rm
if}& c>0\end{array}\right.$


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