real analysis - Find a sequence of measurable functions defined on a measurable set $E$ that converges everywhere on $E$, but not almost uniformly on $E$.

Find a sequence of measurable functions defined on a measurable set $E$ such that the sequence converges everywhere on $E$, but the sequence does not converge almost uniformly on $E$.

I'm having troubles understanding this. I thought that a sequence of functions $\{f_n\}_{n=1}^\infty \to f$ converges almost uniformly if and only if $f_n\to f$ converges in measure. But that's wrong?

Any help would be welcome.

Answer

In light of Egorov's theorem, your measure space will have to be infinite. A good sequence to keep in mind to check that the finite measure space assumption of a theorem is required is the "moving block" $f_n(x) = \chi_{[n,n+1]}(x)$. This converges pointwise to zero, but it fails to converge in a bunch of other senses, including almost uniformly. (To see that, note that it can't converge uniformly on any set whose complement has measure smaller than $1$).