Since $f_n$ converges almost uniformly, for any $\delta>0$, there exists a measurable set $E_\delta$ with measure less than $\delta$ such that $f_n$ converges uniformly on $E_\delta^c$. Thus there are $E_k$ with $\mu(E_k)<1/k$ such that $f_n$ converges uniformly on $E_k^c$.
Let $F_n=\bigcap_{k=1}^nE_k$. Then
$$
\mu(F_n)\leqslant \mu(E_n)<1/n\quad\text{and so}\quad \lim_{n\to\infty}\mu(F_n)=0
$$
Since $F_1\supset\cdots\supset F_n\supset\cdots$, let $F=\bigcap_{n=1}^{\infty}E_n$ and by Monotone class theorem
$$
\mu(F)=\mu\left(\bigcap_{n=1}^{\infty}E_n\right)=\lim_{n\to\infty}\mu(F_n)=0
$$
For any $x\in F^c=\bigcup_{n=1}^{\infty}E_n^c$, there is a $N$ that $x\in E_N^c$. Since $f_n$ converges uniformly on $E_N^c$, $f_n$ converges pointwise on $E_N^c$. This proves that $f_n$ converges pointwise a.e. Also $f_n$ converges in measure since $f_n$ converges pointwise a.e.