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{\bf Question}

If $f:{\bf R^n}\rightarrow{\bf R^*}$ is integrable, show that, for
any $\epsilon>0$,

$m\{x||f(x)|\geq\epsilon\}<+\infty$

Is it necessarily true that $m(\{x||f(x)|>0\}<+\infty$?


\vspace{0.25in}

{\bf Answer}

$\{x||f(x)|\geq\epsilon\}=\{x|f(x)\geq\epsilon\}\cup\{x|f(x)<-\epsilon\}$

Suppose $m(\{x|f(x)|\geq\epsilon\})=+\infty$

Then $\int f_+>\epsilon_x+\infty=+\infty$ and so $f$ is not
integrable.

Let $f(x)=e^{-|x|}$.  Then $\ds \int e^{-|x|}=2\int_0^\infty
e^{-x}=2[-e^{-x}]_0^\infty=2<\infty$

$\{x|e^{-|x|}>0\}={\bf R^0} \,\,\,\, m{\bf R}=+\infty$


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