I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction with the statement in my question.

Let $\Omega\subset \mathbb R^d$ ($d=2,3$) is a bounded Lipschitz domain.

Question:Is it true that for each function $g(x)\in L^2(\Omega)$ one can find a sequence $\{g_n\}_1^\infty$ of $H_0^1(\Omega)$ functions such that $g_n(x)\to g(x),\,a.e$ in $\Omega$ and $|g_n(x)|\leq |g(x)|+\epsilon,\,a.e,\,\forall n\ge 1$ for some $\epsilon>0$ ?

In the case when $g\in L^\infty(\Omega)$ and $m:=\|g\|_{L^\infty(\Omega)}$, then also $g\in L^p(\Omega),\,1\leq p<\infty$ so there is $g_n\in C_0^\infty(\Omega)$ s.t $g_n\to g$ in $L^p(\Omega)$ and from it we can extract a subsequence $g_{n_k}(x)\to g(x),\,a.e$. Finally, we construct the smooth function $\varphi:\mathbb R\to\mathbb R$ s.t $\varphi(t)=t,\,|t|\leq m+1$, $\varphi(t)=m+2,\,t>m+2$, $\varphi(t)=-m-2,\,t<-m-2$ and take the functions $\varphi\circ g_{n_k}$. These functions satisfy $\varphi\circ g_{n_k}(x)\to g(x),\,a.e$ and $|\varphi\circ g_{n_k}(x)|\leq m+2$.

**Remark:** It is easy to show (see for example Th. 4.9 in Brezis' book *Functional Analysis*) that for $g_n\to g$ in $L^p(\Omega)$ there exists a subsequence $g_{n_k}$ s.t $g_{n_k}(x)\to g(x),\,a.e$ and $|g_{n_k}(x)|\leq h(x)$ for some $h\in L^p(\Omega)$. As $C_0^\infty(\Omega)\subset H_0^1(\Omega)$ is dense in $L^2(\Omega)$ we can find $g_n\to g$ in $L^2(\Omega)$ and apply Th. 4.9, but then we get only $|g_{n_k}(x)|\leq h(x)$