Corollary 2.5.5DF54B49EC7D4

Proof. Let $B$ be an open ball with $\overline B\subset \Omega $, and fix $\theta \in C_ c^\infty (B)$ with $\int \theta \, \mathrm dx=1$, for instance a translate of $\rho _ s$ with $s>0$ small. Since $\overline B$ is compact, $\overline B\subset \Omega _\varepsilon $ for all small $\varepsilon $. By proposition 2.5.3(1), the smooth function $T_\varepsilon $ has gradient $((\partial _ jT)_\varepsilon )_ j=0$ on $\Omega _\varepsilon $, so it is equal to a constant $c_\varepsilon $ on the convex set $B$. By proposition 2.5.3(2), $c_\varepsilon =\int T_\varepsilon \theta \, \mathrm dx\to \langle T,\theta \rangle =:c_ B$, and therefore, for every $\chi \in C_ c^\infty (B)$,

If two such balls $B$ and $B'$ intersect, testing with a nonnegative function $\chi \neq 0$ in $C_ c^\infty (B\cap B')$ gives $c_ B=c_{B'}$. Any two such balls containing a point $x\in \Omega $ intersect, so $c(x)=c_ B$, for any such ball $B$ containing $x$, is a well-defined function on $\Omega $; it is locally constant, since $c=c_ B$ on $B$. Every test function $\chi $ has compact support covered by finitely many such balls $B_ l$, and by lemma 2.5.2 it is a sum of test functions $\chi _ l$ supported in $B_ l$. Hence $\langle T,\chi \rangle =\sum _ lc_{B_ l}\int \chi _ l\, \mathrm dx=\int c\chi \, \mathrm dx$. Finally, if $\Omega $ is connected, the locally constant function $c$ is constant, because for each value $a$ the set $\{ c=a\} $ is open and so is its complement. $\square$