Why should the set have finite measure in the following proposition?

Question

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?

Answer 1

Just try $E=\mathbb{R}$, $f_n(x) = x/n$, $f\equiv 0$ for a counterexample.