Here is a proposition in Royden: Assume $E$ has finite measure. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise a.e. on $E$ to $f$ and $f$ is finite a.e. on $E$. Then $\{f_n\}\rightarrow f$ in measure on $E$.
I get the proof, but why doesn't it hold for $E'$ which is of infinite measure?