PomeLemma 8.10.3.A82044827575 Every bounded canonical admissible potential is continuous on the complement of an open set of arbitrarily small $\operatorname {Cap}_1$. The same open set can be chosen for any countable family of such potentials.
Lemma 8.10.3A82044827575
Proof. Choose smooth admissible $u_ j\downarrow u$. By the preceding lemma these functions converge weakly to $u$ in $H^1$. Mazur's lemma gives finite convex combinations $w_ N$ of terms with $j\geq N$ such that
\[ \| w_ N-u\| _{H^1}\leq 8^{-N}. \]
Because $u\leq w_ N\leq u_ N$, they converge pointwise to the canonical function $u$, not just to an unspecified almost-everywhere representative. Put
\[ G_{N_0}=\bigcup _{N\geq N_0} \{ |w_{N+1}-w_ N|>2^{-N}\} . \]
The capacity tools imply
\[ \operatorname {Cap}_1(G_{N_0}) \leq \sum _{N\geq N_0}4^ N\| w_{N+1}-w_ N\| _{H^1}^2 \leq C\sum _{N\geq N_0}16^{-N}\longrightarrow 0. \]
Outside $G_{N_0}$, the successive differences form a uniformly summable series. Thus $w_ N$ converge uniformly there and their limit $u$ is continuous on that closed set. For a countable family, choose the exceptional open sets with capacities less than $\epsilon 2^{-i}$ and take their union. Countable subadditivity proves the last assertion. $\square$
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