Theorem 4.2.22730413AAC920

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. $\square$