@TedShifrin Let $(\Sigma, \mathcal{A}, \mu)$ be a probability space. Let $(A_n)_n$ be a sequence in $\mathcal{A}$. Let $f: \Sigma \rightarrow \mathbb{B}$ be measurable such that $\int_{\Sigma} |\mathbb{1}_{A_n} - f| d\mu \rightarrow 0$ as $n \rightarrow \infty$. Prove there exists $A \in \mathcal{A}$ for which $f = \mathbb{1}_A$ a.s.

The claim is easy if $A_n \searrow A$, or $A_n \nearrow A$. But if, say, $A_{2n} = [-2, 1]$, and $A_{2n+1} = [-1, 2]$, then I don't even see a set $A$ that satisfies the requirement $\mathbb{P}(A_n \triangle A) < \epsilon$ for any $n$.