I'm looking for a counterexample - where a sequence converges pointwise and in measure, but not almost uniformly.
There is a theorem (which I don't know how to prove) which states that if $|f_n| \leq g \in L^1$ and $f_n \rightarrow f$ pointwise and in L^1, then $f_n \rightarrow f$ almost uniformly.
$f_n \rightarrow f$ in $L^1$ implies in measure. I want to abandon $L^1$ convergence and only focus on convergence in measure. Can anyone advise me to the end of this proof, or show me an example of a sequence which converges almost everywhere, in measure, but not almost uniformly?