Theorem 4.4.23DEAF5030EE5E

Proof. Apply $\arctan $ to handle extended limits, and choose intervals $J_ j$ shrinking to the finite transformed limit. There are fine neighborhoods $V_ j$ on which $g\in J_ j$ along $E\setminus \{ a\} $. If infinitely many $V_ j$ are Euclidean neighborhoods the ordinary limit already exists. Otherwise discard finitely many and use amplified positive barriers $v_ j$ for their thin complements on a common ball. Choose $\alpha _ j>0$ with $\sum \alpha _ jv_ j(a)<\infty $. Put $F_ j=(B\setminus V_ j)\cap \{ \alpha _ jv_ j>j\} $ and $F'=\bigcup F_ j$. The sum $v=\sum \alpha _ jv_ j$ tends to infinity along $F'$ at $a$: terms with $j\ge p$ are already above $p$, while each of the finitely many remaining terms is above $p$ sufficiently close to $a$ on its complement. Thus $F'$ is thin. Let $V$ be the fine interior of its complement. Choose an actual Euclidean neighborhood $O_ j$ of $a$ with $O_ j\setminus V_ j\subset \{ \alpha _ jv_ j>j\} $. Then $O_ j\cap V\subset V_ j$, proving the ordinary limit on $E\cap V$.

For the second assertion choose an amplified $u\ge 0$, finite at $a$, infinite along $F$ approaching $a$. If $\liminf _{y\to a,\, y\notin F}(v+u)(y)=\infty $, fine continuity of $u$ implies the fine limit of $v$ is infinity. Otherwise call this liminf $\lambda $ and take $k>\lambda $. Near $a$, $u>k$ on $F\setminus \{ a\} $. Paste $\min (v+u,k)$ off $F$ to the value $k$ on $F\setminus \{ a\} $. The boundary lower limits justify superharmonic pasting. It is bounded below and extends across the polar singleton, with canonical value $\lambda $ at $a$. Its fine neighborhood $\{ w<k\} $ contains $a$; on its complement-of-$F$ portion $w=v+u$. Fine continuity of $w,u$ therefore gives the finite fine limit $\lambda -u(a)$ for $v$. $\square$