In Folland's Real Analysis: Modern Techniques and Their Applications, the following definition is given for a sequence of functions converging in measure.
We say that a sequence $\{f_n\}$ of measurable complex-valued functions on $(X,\mathcal{M}, \mu)$... converges in measure to $f$ if for every $\epsilon>0$, $$\mu(\{x: |f_n(x)-f(x)|\ge\epsilon\})\rightarrow 0 \text{ as } n\rightarrow \infty.$$
My question is, how exactly is the set given in the definition measurable? I see that the set satisfies $$\{x\in X:|f_n(x)-f(x)|\ge\epsilon\} = \Big\{x\in X: x\in f_n^{-1}\Big(B(f(x),\epsilon)^c\Big)\Big\},$$
where $B(f(x),\epsilon)$ denotes an open ball of center $f(x)$ and radius $\epsilon$, and superscript $^c$ denotes the complement of a set. Since each $f_n$ is measurable and the set $B(f(x),\epsilon)^c$ is closed, I know any $f_n^{-1}\Big(B(f(x),\epsilon)^c\Big)$ is measurable. I'm not sure how to proceed from there to show the given set is measurable. Any tips? Thank you for any guidance you have!