This problem is related to Convergence in measure to zero with certain conditions.
(Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable.
For $c_n>0$ such that either $\lim_{n\to \infty}c_n=0$, or $c_n\geq c>0$ for all $n$, and measurable sets $E_n$ with $m(E_n)>0$ consider the sequence $f_n(x):=c_n\mathcal{X}_{E_n}(x).$ )
For the same sequence(same assumptions!) $\{f_n\}_{n\in\mathbb{N}}$, $f_n$ converges almost uniformly to zero, iff $c_n\to 0$ as $n\to \infty$ or $m(\cup_{n\geq N}E_n)\to 0$ as $N\to \infty.$
My approach: ($\Rightarrow$) By definition of almost uniformly, we have that for all $\epsilon>0$ there exists $A_\epsilon $ such that $m(A_\epsilon)<\epsilon$ such that $f_n$ converges to uniformly to $0$ on $A_\epsilon^c$. Observe that we have $A_\epsilon \subset \cup_{n\geq N}E_n$, so $\epsilon\geq m(A_\epsilon)\leq m(\cup_{n\geq N}E_n)$. Now if $x\in A_\epsilon$ then clearly $m(\cup_{n\geq N}E_n)\to 0$ as $N\to \infty$ and if $x\notin A_\epsilon$ then again clearly as $f_n(x)=c_n\mathcal{X}_{E_n}(x)\to 0$, $c_n\to 0.$
($\Leftarrow$) I think to prove this direction is hard.
Any help, comments and suggestions will be appreciated.