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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?

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

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    Yes, and if you go through the proof and see where it goes wrong, it is exactly in the proof of Egoroff's theorem, where you use that a decreasing sequence of measurable sets $(E_n)$ with $\bigcap E_n = \emptyset$ satisfies $\lim \mu(E_n) = 0$. This only holds if you know that the $E_n$ have finite measure (eventually).2012-10-21