Lemma 3.4.22 (Choquet's lemma)B37EED9280E3

Proof. Call $B(q,s)$ a rational ball if $q\in \mathbb Q^ d$ and $s$ is a positive rational number. Enumerate the rational balls $B$ with $\overline B\subset \Omega $ as $B_1,B_2,\ldots $, in such a way that each of them occurs infinitely often. Since $F$ is bounded above on the compact set $\overline{B_ j}$, we have $\sup _{B_ j}F<\infty $. If $\sup _{B_ j}F>-\infty $, choose $v_ j\in \mathcal F$ with $\sup _{B_ j}v_ j>\sup _{B_ j}F-1/j$; otherwise let $v_ j$ be any member of $\mathcal F$. Put $v=\sup _ jv_ j\leq F$. If $B$ is a rational ball with $\overline B\subset \Omega $, then $B=B_ j$ for infinitely many $j$, so $\sup _ Bv=\sup _ BF$. Every ball $B(x,r)$ contains a rational ball that contains $x$: choose $q\in \mathbb Q^ d$ with $|q-x|<r/3$ and a rational number $s\in (r/3,2r/3)$; then $x\in B(q,s)\subset B(x,r)$. Conversely, a rational ball $B(q,s)$ containing $x$ contains $B(x,s-|q-x|)$. Hence $F^*(x)$ is the infimum of $\sup _ BF$ over the rational balls $B\ni x$ with $\overline B\subset \Omega $, and similarly for $v^*(x)$. Therefore $v^*=F^*$. $\square$