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. Thenare test functions and $u_j\to u$: for each fixed test function $\varphi$, the cutoff is eventually one near its support, and the reflected mollifier approximates $\varphi$ 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 ${ϵ}_j\to 0$ 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 $u$.