4.2.21 Countable covers and quasi-open setsA51B9EF16CF8

Proof. We first reduce to sets of a special form attached to balls with rational centers and radii; for each such ball, Choquet's lemma then selects countably many of these sets. Let $U=\bigcup _{i\in I}U_ i$. Call a set basic if it has the form $Q\cap \{ u>a\} $, where $Q$ is a ball with rational center and rational radius such that $\overline Q\subset \Omega $, $u$ is subharmonic on an open neighborhood of $\overline Q$ in $\Omega $, and $a\in \mathbb R$. Basic sets are finely open. Every point $x\in U$ lies in a basic set contained in some $U_ i$. Indeed, if $x\in U_ i$, then proposition 4.2.4 provides a Euclidean open set $V$, a function $u\in \mathrm{SH}(V)$ and $a<u(x)$ with $\{ y\in V:u(y)>a\} \subset U_ i$, and we choose a rational ball $Q$ with $x\in Q$ and $\overline Q\subset V$. Let $\mathcal B$ be the family of basic sets contained in some $U_ i$; thus $U=\bigcup \mathcal B$.

Fix a rational ball $Q$, and let $\mathcal B_ Q$ consist of the members of $\mathcal B$ of the form $Q\cap \{ u>a\} $; we may assume that $\mathcal B_ Q$ is nonempty. For every pair $(u,a)$ as in the definition of basic sets with $Q\cap \{ u>a\} \in \mathcal B_ Q$, put

The supremum is less than $+\infty $, because $u$ is upper semicontinuous on a neighborhood of the compact set $\overline Q$ and hence bounded above on $\overline Q$. Thus $v$ is subharmonic on $Q$, $v\leq 1$, and $\{ v>0\} =Q\cap \{ u>a\} $.

Let $F$ be the supremum of all the functions $v$ so obtained. By Choquet's lemma (lemma 3.4.22), there is a sequence $(v_ k)$ among them such that $F'=\sup _ kv_ k$ satisfies $(F')^*=F^*$. By part (2) of theorem 4.1.26, the sets $\{ F\neq F^*\} $ and $\{ F'\neq (F')^*\} $ are polar; let $P_ Q$ be their union. If $y\in Q\setminus P_ Q$ and $v(y)>0$ for some $v$, then $F'(y)=(F')^*(y)=F^*(y)=F(y)>0$, so $v_ k(y)>0$ for some $k$. Hence the union of $\mathcal B_ Q$ is contained in the union of $P_ Q$ and the countably many basic sets $\{ v_ k>0\} $.

Doing this for each of the countably many rational balls $Q$, we obtain countably many members of $\mathcal B$ whose union covers $U$ outside the set $P=\bigcup _ QP_ Q$, which is polar by proposition 4.1.11. For each selected member choose one index $i$ such that the member is contained in $U_ i$. The set $J$ of chosen indices is at most countable, and $U\setminus \bigcup _{j\in J}U_ j\subset P$. This proves the first assertion.

For the second, let $U$ be finely open, and apply the first assertion to the family of all basic sets contained in $U$, whose union is $U$ by proposition 4.2.4. We obtain countably many basic sets whose union $V\subset U$ is such that $U\setminus V$ is polar. Each basic set is Borel, since subharmonic functions are Borel measurable, so $V$ is Borel.

For the measurability assertions, it suffices to show that the polar set $U\setminus V$ is contained in a Borel set $N$ that is null for Lebesgue measure and for every finite positive measure of compact support with finite energy. Since $\mathbb R^ d$ has a countable base, $U\setminus V$ is covered by countably many connected open sets $V_ k$ carrying nontrivial subharmonic functions $\psi _ k$ with $(U\setminus V)\cap V_ k\subset \{ \psi _ k=-\infty \} $. The Borel set $N=\bigcup _ k\{ \psi _ k=-\infty \} $ is polar by proposition 4.1.11; it is Lebesgue null by proposition 3.4.8, and null for measures of finite energy by theorem 4.1.17. $\square$

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. $\square$