Let $\Omega$ be a subset of either $\mathbb{R}^n$ ($n\geq 3$, if it matters) or of a compact manifold. In either case, we'll call the manifold $M$. Let $V_i\subset V_{i+1} \subset \Omega$ be an increasing sequence of subsets. Assume that $\cup V_i = \Omega$ up to a set of measure zero. Let $$A(\Omega) = \{f\in W^{1,2}_c(M): f|_{M\setminus \Omega} = 0\},$$ or, in other words, $A(\Omega)$ are the $W^{1,2}$ functions with compact support that are zero except on $\Omega$.
I want to know under what conditions $\cup A(V_i)$ is dense in $A(\Omega)$. If it is easier, we can assume $\Omega$ and/or the $V_i$ are open, or similar, but I am interested in the case where they may only be measurable.
If $\Omega$ is open and the $V_i$ are $\Omega\setminus B_{1/i}$, where $B_{1/i}$ is a ball of radius $1/i$, then it is easy to show that $\cup A(V_i)$ is dense in $A(\Omega)$ by using a cutoff function. The same kind of argument likely works for general $V_i$ that are $\Omega$ minus some open set with "nice" boundary, but I'm not sure about anything more general than that.
