Let $\Omega\subset\mathbb{R}^m$ be a domain with smooth boundary. Let $\psi:\Omega\to\mathbb{R}$ be a function of bounded variation. Let $D\subset\Omega$ be a countable dense subset such that $\psi$ is continuous at all $x\in D$. There is a sequence of smooth functions $\{f_n\}$ such that as $n\to\infty$,$\|f_n-\psi\|_{L^2(\Omega)} = 0$. It is also given that $\forall x\in D$ as $n\to\infty$,$f_n(x)\to\psi(x)$.

I want to prove that $f_n\to \psi$ almost everywhere. Appreciate any hints/help.

**Edit:** (after answer by Christian Remling, and after a close vote)

What if Fourier transform of every $f_n$ has a compact support? to make this scenario possible, Assume domain $\Omega$ is $\mathbb{T}^m$ and $\{f_n\}$ are all trigonometric polynomials of various degrees. In this case, will the $L^2$ convergence imply almost everywhere convergence, for a bounded variation function $\psi:\mathbb{T}^m\to\mathbb{R}$ ?