I am trying to prove the following:
Let $\left(f_n\right)$ be a sequence of measurable functions on a measure space $\left( X, \mathcal{A}, \mu \right)$, and let $f: X \to \mathbb{R}$. Suppose that $$\lim_{n \to \infty} \mu\left( \{ x \in X : \vert f_n(x) - f(x) \vert \geq \varepsilon \} \right) = 0,$$ for all $\varepsilon>0$. Prove that $f$ is measurable. I tried to imitate the proof that pointwise limit of measurable functions is measurable, but got stuck because I can only guarantee "convergence" on a set whose complement has arbitrarily small measure (but is not a null set). Any hints?