Let $f_n$, $f:(X,\mu)\to\mathbf{C}$ be measurable. We say $f_n\to f$ almost uniformly if for every $\varepsilon>0$, there is a measurable $A\subseteq X$ such that $\mu(A)<\varepsilon$ and $f_n\to f$ uniformly on $A^c$.
I want to prove or disaprove the following statement:
Let $f_n$, $f:(X,\mu)\to\mathbf{C}$ be measurable. Then $f_n\to f$ almost uniformly on $X$ if and only if there is $\{\varepsilon_n\}\downarrow 0$ such that $\lim_{k\to\infty}\mu(\bigcup_{n\geq k}\{|f_n-f|\geq\varepsilon_n\})=0$
The if part is true: Assume there is $\{\varepsilon_n\}\downarrow 0$ such that $\lim_{k\to\infty}\mu(\bigcup_{n\geq k}\{|f_n-f|\geq\varepsilon_n\})=0$. Then given $\varepsilon>0$, there is $k_\varepsilon\in\mathbb{N}$ such that $$ \mu(B_\varepsilon)<\varepsilon,\quad B_\varepsilon=\bigcup_{n\geq k_\varepsilon}\left\{|f_n-f|\geq \varepsilon_n\right\}. $$ Then $|f_n-f|<\varepsilon_n$ on $B_\varepsilon^c$ for all $n\geq k_\varepsilon$. Let $n\to\infty$ to see that $f_n\to f$ uniformly on $B_\varepsilon^c$.
However, I don't know how to prove the only if part, and can't find a counterexample either.