= Density of test functions in distributions
Every <distribution> on an <open set> $U$ is a limit of <test functions> in <weak convergence of distributions>. On the whole space $\mathbb R^n$, choose a <mollifier> $\rho$ and compactly supported smooth cutoffs $\chi_j$ equal to one on expanding balls. Then
$$
u_j=\chi_j(u*\rho_{1/j})
$$
are <test functions> and $u_j\to u$: for each fixed <test function> $\phi$, the cutoff is eventually one near its <support>, and the reflected mollifier approximates $\phi$ with every derivative in one common <compact set>.
For a general <open set> $U$, ordinary fixed-radius <convolution> need not be defined near its boundary. Choose an exhaustion $K_j$ of $U$ and <cutoff functions> $\chi_j$ compactly supported inside $U$, equal to one near $K_j$. Extend the <compactly supported distribution> $\chi_j u$ by zero to the whole space, then convolve with a mollifier of radius $\epsilon_j\to0$ small enough that the resulting support stays inside $U$. For every fixed test, the cutoff eventually has no effect, so these smooth compactly supported approximants converge to $u$.
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