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{\bf Question}

Find an example of a function $f:{\bf R}\rightarrow{\bf R}$ with
the properties

\begin{itemize}
\item[i)]
$f\not\epsilon R[0,1]$

\item[ii)]
$f\epsilon L[0,1]$

\item[iii)]
$|f|\epsilon R[0,1]$
\end{itemize}

Is it true that if $|f|\epsilon L[0,1]$ then $f\epsilon L[0,1]$?

\vspace{0.25in}

{\bf Answer}

Example $f=\left\{\begin{array}{cl}1&x\epsilon Q\cap[0,1]\\
-1&x\epsilon [0,1]-Q\\ 0&x\not\epsilon [0,1]\end{array}\right.$

Not true, let $E$ be a non-measurable subset of $[0,1]$.

$g=\left\{\begin{array}{cl}1&x\epsilon E\\ -1&x\epsilon [0,1]-E\\
0&x\not\epsilon [0,1]\end{array}\right.$


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