Let $\Omega$ be a bounded domain. I have a function $f_n\colon \Omega \to \mathbb{R}$ such that $f_n \to f$ in $L^2(\Omega)$ and $f \leq 0$ a.e. Let $g_n \in L^2(\Omega)$ be uniformly bounded in $n$.
Define $A_n := \{ x : f_n(x) \geq \frac 1n\}$. So in the limit, $A_n$ should tend to the empty set.
Can I say that
$$n\int_{A_n} g_n(x) \to 0$$
I guess not, but what if I have a rate of convergence for $A_n$ or something like that? as $n \to \infty$?
$n\int_{A_n} g_n \to 0$ for every uniformly bounded sequence $\{g_n\}$ iff $n \mu \{x: f_n(x) \geq \frac 1 n\} \to 0$. [In one direction take $g_n=1$ for all $n$ to prove this].