I want to prove following statement in Folland: If $f_n \to f$ almost uniformly, then $f_n \to f$ a.e. and in measure.
This is what I did: For $k \in \mathbb{N}$, we choose $F_k$ s.t. $\mu(F_k) < \frac{1}{k}$ and $f_n \to f$ uniformly on $F_k^c$ (so I can do this, how can I verify that there exists such $F_k$?). Then, take $E = \bigcup_1^{\infty} F_k$. Then $f_n \to f$ on $E^c$ (is it uniform or not?), and I try to verify $\mu(E) = 0$ (but have problems also), so if this is valid, things are OK for a.e. convergence.
What can I further do for convergence in measure?
This is a homework question, so if you give reasonable hints, I will be very happy. Thanks!