Meyers–Serrin density on an arbitrary open set

Fix the compact exhaustion $K_j$ of [F1] and, using [F2] and Countable Choice, cutoffs ${\psi }_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $0\le {\psi }_j\le 1$, ${\psi }_j=1$ on a neighbourhood of $K_j$ and $\mathrm{supp}{\psi }_j\subset \mathrm{int}{K}_{j+1}$; set ${\chi }_j:=1-{\prod }_{i\le j}(1-{\psi }_i)$ and ${\eta }_1:={\chi }_1$, ${\eta }_j:={\chi }_j-{\chi }_{j-1}$ for $j\ge 2$. Then each ${\eta }_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is nonnegative with $0\le {\eta }_j\le 1$, satisfies $\mathrm{supp}{\eta }_j\subset \mathrm{int}{K}_{j+1}$ and ${\eta }_j=0$ on a neighbourhood of $K_{j-1}$, the supports are locally finite, and ${\sum }_j{\eta }_j=1$ on $\mathrm{\Omega }$; hence ${\sum }_j{\eta }_{j}u=u$ as a locally finite sum of classes. The empty case $\mathrm{\Omega }=\mathrm{\varnothing }$ is trivial because the only class is $0$, so assume $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$.

For each $j$, [F4] gives ${\eta }_{j}u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$, supported in the compact set $\mathrm{supp}{\eta }_j\subset \mathrm{int}{K}_{j+1}$; using [F3] on the open set $U_j:=\mathrm{int}{K}_{j+1}\subset \subset \mathrm{\Omega }$ and [F8] to keep the support inside $\mathrm{\Omega }$, choose ${\epsilon }_j>0$ so small that $v_j:={\rho }_{{\epsilon }_j}\ast ({\eta }_{j}u)$ is a smooth function compactly supported in $U_j$ and $\mathrm{\parallel }{v}_j-{\eta }_{j}u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}=\mathrm{\parallel }{v}_j-{\eta }_{j}u{\mathrm{\parallel }}_{W^{k,p}(U_j)}<\delta 2^{-j-1}\mathrm{.}$ For $j\ge 2$, also take ${\epsilon }_j<\frac{1}{2}\mathrm{dist}(\mathrm{supp}{\eta }_j,K_{j-1})>0$; then $v_j$ vanishes near $K_{j-1}$, so the mollified pieces remain locally finite.

Define $v:={\sum }_j{v}_j$ and $e_j:=v_j-{\eta }_{j}u$. Since the $v_j$ have locally finite supports, $v$ is a locally finite sum of smooth compactly supported functions on $\mathrm{\Omega }$, hence $v\in C^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})$; and $v-u={\sum }_j(v_j-{\eta }_{j}u)={\sum }_j{e}_j$ as a locally finite sum, with each $e_j\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$ and $\mathrm{\parallel }{e}_j{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}<\delta 2^{-j-1}$ by step 2.1.

For every $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ one has $D^{\alpha }(v-u)={\sum }_j{D}^{\alpha }{e}_j$ as locally integrable classes: near any point of $\mathrm{\Omega }$ only finitely many ${\eta }_j$, hence only finitely many $e_j$, are nonzero, and on that neighbourhood the identity follows from the linearity and locality of weak differentiation applied to the finite sum; the identity therefore holds as an $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega })$ identity.

Let $S_J:={\sum }_{j\le J}{e}_j$, so that $D^{\alpha }{S}_J\to D^{\alpha }(v-u)$ pointwise for every $\mathrm{\mid }\alpha \mathrm{\mid }\le k$ by the local finiteness of step 4.1. Fatou's lemma applied to the nonnegative functions $g_J:={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le k}\mathrm{\mid }{D}^{\alpha }{S}_J{\mathrm{\mid }}^p$ gives ${\int }_{\mathrm{\Omega }}{\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le k}\mathrm{\mid }{D}^{\alpha }(v-u){\mathrm{\mid }}^p\le {\mathrm{lim\hspace{0.17em}inf}}_{J\to \mathrm{\infty }}{\int }_{\mathrm{\Omega }}{g}_J={\mathrm{lim\hspace{0.17em}inf}}_{J\to \mathrm{\infty }}\mathrm{\parallel }{S}_J{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}^p\le ({\sum }_{j=1}^{\mathrm{\infty }}\mathrm{\parallel }{e}_j{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}{)}^p<{\delta }^p,$ where the middle equality is the norm formula of [F6] and the last inequality is the triangle inequality for the norm together with ${\sum }_j\mathrm{\parallel }{e}_j{\mathrm{\parallel }}_{W^{k,p}}<{\sum }_j\delta 2^{-j-1}\le \delta$.

By step 5.1, $\mathrm{\parallel }v-u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}<\delta$; by step 3.1, $v\in C^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})$; and $v=(v-u)+u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$ because $v-u$ and $u$ both are. Since $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$ and $\delta >0$ were arbitrary, $C^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})\cap W^{k,p}(\mathrm{\Omega };\mathbb{K})$ is dense.