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{\bf Question}

A function $f:{\bf R^n}\rightarrow {\bf R}$ is said to be
measurable iff

\begin{itemize}
\item[i)]
for all $c\epsilon{\bf R} \,\,\, \{x|f(x)\leq c\}$ is measurable.
\end{itemize}

show that statement (i) is equivalent to each of the statements
below

\begin{itemize}
\item[ii)]
for all $c\epsilon{\bf R} \,\,\, \{x|f(x)<c\}$ is measurable.

\item[iii)]
for all $c\epsilon{\bf R} \,\,\, \{x|f(x)\geq c\}$ is measurable.

\item[iv)]
for all $c\epsilon{\bf R} \,\,\, \{x|f(x)>c\}$ is measurable.
\end{itemize}

\vspace{0.25in}

{\bf Answer}

$\ds \{x|f(x)<c\}=\bigcup_{n=1}^\infty\{x|f(x)\leq
c-\frac{1}{n}\}$

${}$

$\ds \{x|f(x)\geq c\}={\bf R^n}-\{x|f(x)<c\}$

${}$

$\ds \{x|f(x)>c\}=\bigcap_{n=1}^\infty\{x|f(x)\geq
c+\frac{1}{n}\}$



\end{document}