Lemma 2.5.2677586F8712C

Proof. For every $x\in K$ choose an index $j(x)$ with $x\in U_{j(x)}$ and a radius $r_ x>0$ with $\overline{B(x,r_ x)}\subset U_{j(x)}$. Finitely many of the balls $B(x,r_ x)$ cover $K$; let $L_ j$ be the union of the closures of those among them with $j(x)=j$. Then $L_ j$ is a compact, possibly empty, subset of $U_ j$, and $K\subset L_1\cup \cdots \cup L_ n$. By lemma 2.5.1(4), there are smooth cutoff functions $\psi _ j$ for $L_ j$ in $U_ j$ which equal $1$ on neighborhoods $V_ j$ of $L_ j$. Put $\eta _1=\psi _1$ and $\eta _ j=\psi _ j(1-\psi _1)\cdots (1-\psi _{j-1})$ for $2\leq j\leq n$. These functions are smooth with values in $[0,1]$, and $\operatorname{Supp}\eta _ j\subset \operatorname{Supp}\psi _ j$ is a compact subset of $U_ j$. By induction on $n$,

which takes values in $[0,1]$ and equals $1$ on the neighborhood $V_1\cup \cdots \cup V_ n$ of $K$. If $\chi $ is a test function with $\operatorname{Supp}\chi \subset U_1\cup \cdots \cup U_ n$, we apply this to $K=\operatorname{Supp}\chi $; then $\chi =\sum _ j\chi \eta _ j$, and $\operatorname{Supp}(\chi \eta _ j)\subset \operatorname{Supp}\eta _ j$ is a compact subset of $U_ j$. Finally, let a distribution $T$ vanish on each member of a family of open subsets of $\Omega $, and let $\chi $ be a test function supported in their union. The compact set $\operatorname{Supp}\chi $ is covered by finitely many members $U_1,\ldots ,U_ n$ of the family, and the decomposition $\chi =\sum _ j\chi \eta _ j$ just obtained gives $\langle T,\chi \rangle =\sum _ j\langle T,\chi \eta _ j\rangle =0$. $\square$