Let $\Omega \subset \mathbb R^N$ then for a given compact subset $E \subset \Omega$ define the set: $\mathcal K_E:=\{v\in W^{1,2}_0(\Omega) :v\ge 1 \;on\;E\;in\;W^{1,2}(\Omega)\}\;$.
Take $0\le \zeta \in C^1_0(\Omega)$ and let $\mu$ denote a nonnegative Radon measure on $\Omega$. Assume $\mu(E) \le C {\vert \vert \nabla \zeta \vert \vert}_{L^2(\Omega)}$ for some positive constant $C \gt 0$
Since $C^1_0(\Omega) \cap \mathcal K_E$ is dense in $\mathcal K_E$, $\mu(E) \le C \;{\inf}_{ \zeta \in \mathcal K_E} {\vert \vert \nabla \zeta \vert \vert}_{L^2(\Omega)}$
I have trouble understanding the last sentence. I don't see why density is sufficient in order to take the infimum over all $\zeta \in \mathcal K_E$ and I suppose I'm missing some Propositions here.
I would appreciate any help.
Thnaks in advance