<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt]{article}
\newcommand{\ds}{\displaystyle}
\parindent=0pt
\begin{document}

{\bf Question}

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


\vspace{0.25in}

{\bf Answer}

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


\end{document}
</pre></body></html>