Proposition 4.2.23F3EA8F8D8D8F

Proof. We construct $O$ as the union of an open set covering the polar part of $U$ near $L$ and of open sets outside which the functions defining the rest of $U$ are continuous on $L$. By the second assertion of theorem 4.2.22, there are at most countably many sets $V_ k=Q_ k\cap \{ u_ k>a_ k\} \subset U$, with $Q_ k$, $u_ k$ and $a_ k$ as there, whose union $V$ is such that $U\setminus V$ is contained in a polar set $P$. Discarding the empty ones, we may assume that every $V_ k$ is nonempty and that $k$ ranges over a set $\mathcal K\subset \{ 1,2,\ldots \} $. Since $V_ k\neq \varnothing $, we may assume that $u_ k$ is defined on an open ball $Q_ k'\supset \overline{Q_ k}$ and is nontrivial there; by proposition 3.4.8, it is then locally integrable, hence not identically $-\infty $ on any ball in $Q_ k'$. Fix an open set $W_0$ with $L\subset W_0\Subset B$.

First we cover the polar part. The set $P\cap W_0$ is polar and its closure is a compact subset of $B$, so $C_ B^*(P\cap W_0)=0$ by corollary 4.1.24. Hence there is an open set $O_0$ with $P\cap W_0\subset O_0\subset W_0$ and $C_ B(O_0)<\eta /2$.

Next we deal with the discontinuities of the functions $u_ k$. For each $k\in \mathcal K$, the compact set $\overline{Q_ k}\cap L$ lies in the open set $Q_ k'\cap B$. Each of its points therefore lies in a ball $D'$ such that $D'\Subset D\Subset Q_ k'\cap B$ for another ball $D$, and finitely many such balls $D'$ cover $\overline{Q_ k}\cap L$. Enumerate all the pairs of balls obtained in this way, for all $k\in \mathcal K$, as $(D'_ j,D_ j)$ with $j$ in a set $\mathcal J$ of positive integers, and let $k(j)$ be the index $k$ for which the pair $(D'_ j,D_ j)$ was chosen. For each $j\in \mathcal J$, the quasi-continuity theorem (theorem 4.1.30), applied to $u_{k(j)}$ on $Q_{k(j)}'$ with the ball $D_ j\Subset Q_{k(j)}'$ and the compact set $\overline{D'_ j}\cap L\subset D_ j$, provides an open set $O_ j\Subset D_ j$ with $C_{D_ j}(O_ j)<\eta 2^{-j-1}$ such that $u_{k(j)}$ is finite and continuous on $(\overline{D'_ j}\cap L)\setminus O_ j$. By proposition 4.2.12, $C_ B(O_ j)=C_ B^*(O_ j)\leq C_{D_ j}^*(O_ j)=C_{D_ j}(O_ j)<\eta 2^{-j-1}$.

Put $O=O_0\cup \bigcup _{j\in \mathcal J}(O_ j\cap W_0)$. This is an open subset of $W_0$, so $O\Subset B$, and countable subadditivity and monotonicity of the outer capacity (theorem 4.1.19) give

It remains to show that $U\cap F$ is relatively open in $F=L\setminus O$. Since $L\subset W_0$, the set $(U\setminus V)\cap F$ is contained in $(P\cap W_0)\setminus O_0=\varnothing $, so $U\cap F=V\cap F$. Let $y\in V\cap F$. Then $y\in V_ k$ for some $k\in \mathcal K$; since $y\in \overline{Q_ k}\cap L$, we have $y\in D'_ j$ for some $j$ with $k(j)=k$. Since $L\subset W_0$, we have $O_ j\cap L\subset O$, so $D'_ j\cap F\subset (\overline{D'_ j}\cap L)\setminus O_ j$, and the function $u_ k$ is finite and continuous on $D'_ j\cap F$. Hence

is relatively open in $F$, contains $y$, and is contained in $V_ k\cap F$. Therefore $U\cap F=V\cap F$ is relatively open in $F$.

Write $U\cap F=T\cap F$ with $T$ open in $\mathbb R^ d$. Then $(U\cup O)\cap L=(U\cap F)\cup (O\cap L)=(T\cup O)\cap L$ is relatively open in $L$. Conversely, if $(U\cup O)\cap L$ is relatively open in $L$, then its intersection with $F$, which is $U\cap F$, is relatively open in $F$. This proves the equivalence of the two formulations. $\square$