Consider $\Omega \subset \mathbb{R}^n$ compact. I know that there exists $u\in W^{1,\infty}(\Omega)$ such that
$$\sup_{\phi \in W^{1,\infty}(\Omega)} \int_\Omega \phi \,d \mu = \int_\Omega u \,d\mu $$
I want to prove that $\int_\Omega u \, d\mu \leq C$ for a certain (prefixed) real constant $C.$
I was able to prove that for arbitrary $v\in C^1(\Omega)$ we have
$$\int_\Omega v \, d\mu \leq C $$ Now seems clear the result follows by approximation. Take a sequence $v_k \in C^1(\Omega)$ converging to $u$ in $W^{1,\infty}(\Omega)$. In Santambrogio's book (Optimal Transport for applied mathematicians. pg 131) there's a hint to take $v_k = \eta_k\star u$
My problem: All the result I know for approximations via convolutions require $p<\infty$ and here I'm dealing exactly with $p=\infty$. Any ideas how this argument would work?