Let $\mu$ be a positive Radon measure on a bounded domain $\Omega$ such that $\mu \in H^{-1}(\Omega)$. Furthermore, $\mu$ has support in $Z \subset \Omega$, which is a subdomain of $\Omega$.
Suppose I have a function $z \in H^1_0(\Omega)$ such that $z=0$ a.e. in $Z$.
Is it enough to conclude that $$\langle \mu, z \rangle_{H^{-1}, H^1_0} = 0?$$
I just a saw thread (Radon measure with support in a set) related to this... but I think mine is different.
No, it is not enough. Your $\mu$ could be a $d-1$-dimensional measure on a hyperplane, e.g., a line measure in two dimensions. Then, the support $Z$ is the hyperplane, which has measure zero. Hence, all function vanish a.e. on this hyperplane.