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{\bf Question}

Suppose $\{f_n\}$ is a sequence of functions each of which is
finite a.e.  Show that, for almost all $x$ in ${\bf R^n}$,
$f_n(x)$ is finite for all $n$.



\vspace{0.25in}

{\bf Answer}

Let $A_n=\{x|f_n(x)=+\infty\vee f_n(x)=-\infty\}$

Then $m(A_n)=0$.  Let $S=\{x|$ for all $n\epsilon{\bf N}\,\,\,
-\infty&lt;f_n(x)&lt;\infty\}$

Then $\ds S={\bf R^n}-\bigcup_{n=1}^\infty A_n$

Therefore $\ds CS=\bigcup_{n=1}^\infty A_n$ and $m(CS)=0$


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