Let Ω be a bounded domain in $\mathbf R^N$. It is well-known that $$ \forall u\in L_1(\Omega),\;∃(u_n)_n\subset C_0^\infty(Ω)\text{ s.t. } u_n\to u\text{ in }L_1(\Omega)\text{ as } n\to\infty. $$ I would like to extend this type of result to $C(\overline{\Omega})^\ast$, which is slightly larger than $L_1(\Omega)$. More precisely, I would like to know whether it is possible to prove that $$ \forall u\in C(\overline{\Omega})^\ast, ∃(u_n)_n\subset C_0^\infty(Ω) \text{ s.t. } \sup_{n∈N}\|u_n\|_{L_1(\Omega)}<+\infty \text{ and } u_n\to u \text{ $\ast$-weakly in } C(\overline{\Omega})^\ast $$ This statement is true if $u=\delta \in C(\overline{\Omega})^*$ because it is well-known that $\delta$ can be approximated by mollifier. But how about when $u$ is in $C(\overline{\Omega})^\ast$ and not Dirac delta function?
I would appreciate it if you could tell me comments or advises.