In Real Analysis, Egoroff's Theorem stated that:
"Assume $E$ has finite measure. Let ${f_n}$ be a sequence of measurable functions on $E$ that converges pointwise on E to the real-valued function $f$. Then for each $\epsilon >0$, there is a closed set F contained in $E$ for which ${f_n} \rightarrow f$ uniformly on $F$ and $m(E \setminus F) < \epsilon$.
I would like to prove that if the domain of functions does not have finite measure, then Egoroff's Theorem does not hold. I tried to find a counter example but I can not.