I have recently proven that $C_0^\infty (\Omega)$ is dense in $L^p (\Omega)$ for $1\leq p < \infty$. It is known that $C_0^\infty (\Omega)$ isnt dense in $L^\infty (\Omega)$. However there is a sequence with following properties for $\Omega \subset \mathbb{R}^d$:
1: $||u_n||_{L^\infty} \leq ||u||_{L^\infty}$
2: $u_n \rightharpoonup^\star u$ in $L^\infty (\Omega)$
3: $u_n \rightarrow u$ almost everywhere in $\Omega$
I got the tip that to approximate $\Omega$ with $\Omega_n = \left\{x\in\Omega|\operatorname{dist}(x, \partial \Omega) > \frac{2}{n}, |x|<n \right\}$.
Now i take the $\xi_n = \chi_{\Omega_n}$ and the Dirac sequence $\rho_n$ and define $$v_n = \rho_n \star (\xi_n u)$$ I have to find a subsequence of that that works with the 3 properties. Can anyone help me find these subsequence?
Thanks in advance