Let $1 \leq p < \infty$ be fixed and let $\Omega \subseteq \mathbb{R}^n$ be open. Let $(Q_n)_{n \in \mathbb{N}}$ be a uniformly bounded family of operators on $L^p(\Omega)$, i.e. there exists $C>0$ such that $\|Q_n\| \leq C$ for all $n \in \mathbb{N}$.

Now suppose that for all $u \in L^p(\Omega)$, we have $Q_n u \longrightarrow u$ pointwise almost everywhere. Does this imply that $Q_n \longrightarrow \mathrm{id}$ in the strong operator topology, i.e. $Q_n u \longrightarrow u$ in $L^p(\Omega)$ for each $u \in L^p(\Omega)$?

I am looking for a proof or a counterexample. Does the answer depend on the choice of $\Omega$?

**Edit:** What if we additionally have $\|Q_n\| \longrightarrow 1$, or even $\|Q_n\| \leq 1+ \frac{C}{n}$ for some $C>0$? I forgot to ask about this additional condition in my first post. I am aware that this probably still does not fix the situation, but I didn't manage to construct an example from the answer given that satisfies this additional requirement.