Capacity-polar sets, quasi-everywhere, and subharmonic polar sets

Let $K\subseteq \mathbb{C}$ be compact; call it the conductor when it is fixed as the ambient set of a quasi-everywhere statement. Capacity is the logarithmic capacity of Robin constant and logarithmic capacity of a compact set, whose conventions $\mathrm{cap}(\mathrm{\varnothing })=0$ and $\mathrm{cap}(F)>0$ or $=0$ for nonempty compact $F$ are used verbatim.

Capacity-polar sets. A set $E\subseteq \mathbb{C}$ is capacity-polar when

$\mathrm{cap}(F)=0\text{for every compact }F\subseteq E\mathrm{.}$

The restriction to compact subsets is deliberate: capacity is defined here for compact sets, so the definition tests $E$ through its compact parts, and a capacity-polar set may be neither compact nor Borel. A compact $F$ is capacity-polar exactly when $\mathrm{cap}(F)=0$, and $\mathrm{\varnothing }$ is capacity-polar by the convention $\mathrm{cap}(\mathrm{\varnothing })=0$. A set $E$ is nonpolar when it is not capacity-polar, that is, when it contains a compact set of positive capacity.

Quasi-everywhere. Let $P$ be a property of the points of a compact conductor $K$, that is, a statement $P(z)$ for $z\in K$. One says that $P$ holds quasi-everywhere on $K$, abbreviated $P$ holds q.e. on $K$, when there is a Borel capacity-polar set $E\subseteq K$ with

$P(z)\text{ holds for every }z\in K\setminus E\mathrm{.}$

The exceptional set $E$ is required to be Borel so that the statement has a measurable exceptional carrier; the definition itself asks nothing about the values of $P$ on $E$. If $P$ holds everywhere on $K$ it holds q.e. on $K$, with $E=\mathrm{\varnothing }$. A set $N\subseteq K$ is called q.e.-negligible when it is contained in a Borel capacity-polar subset of $K$; a property holds q.e. exactly when it fails on a q.e.-negligible set.

Subharmonic polar sets. A set $E\subseteq \mathbb{C}$ is subharmonically polar when for every $x\in E$ there are a complex domain $U_x\ni x$ and a function $u_x$ subharmonic on $U_x$ (Subharmonic functions on plane domains) with

$E\cap U_x\subseteq \{z\in U_x:u_x(z)=-\mathrm{\infty }\}\mathrm{.}$

Here a complex domain is a nonempty connected open set (A complex domain is a nonempty connected open subset of $\mathbb{C}$). Subharmonicity already requires that $u_x$ be not identically $-\mathrm{\infty }$ on $U_x$; the requirement in the literature that the witness be "not identically $-\mathrm{\infty }$" is therefore automatic in this convention. If $E$ is contained in a single complex domain carrying one such witness, the local condition holds with that one function; the definition uses the local form so that unbounded or noncompact $E$ need no global witness.

The two notions are defined independently and are not identified here. Capacity-polar is an inner-capacity condition on compact subsets, while subharmonically polar is a local $-\mathrm{\infty }$-locus condition. Under Dependent Choice, for compact $E$ the two are equivalent (Compact capacity-zero sets and subharmonic minus-infinity loci), and that equivalence is a theorem, not part of this definition. To pass from local witnesses to the global witness in that lemma when $E$ is compact, cover $E$ by finitely many open discs whose closed discs lie in the respective local witness domains. Each compact piece obtained by intersecting $E$ with one of these closed discs has capacity zero by the compact converse in the lemma. Its specified finite-union clause supplies a global subharmonic witness for their union $E$, and its compact converse gives $\mathrm{cap}(E)=0$. The global-to-local direction uses the same witness on each neighbourhood. In particular, no statement here asserts that a capacity-polar set of a compact conductor is the $-\mathrm{\infty }$ locus of one subharmonic function, nor that the definition extends to arbitrary non-Borel sets.

Polarity inherits the empty and inclusion cases. Every subset of a capacity-polar set is capacity-polar, since every compact subset of the subset is a compact subset of the larger set; in particular $\mathrm{\varnothing }$ is capacity-polar, and a set is capacity-polar if and only if all its subsets are. The corresponding statements for subharmonically polar sets hold by restricting the local witnesses.

The diagonal convention is not affected. The exceptional sets here are compared only through capacities and $-\mathrm{\infty }$ loci; no change of the logarithmic kernel on a null set is made or permitted by this definition, and the diagonal value $+\mathrm{\infty }$ of Logarithmic potential and energy of a positive compactly supported measure plays no role.

Choice. No choice principle is used in this definition. "Every compact $F\subseteq E$" is a statement about a fixed collection of sets, the exceptional Borel set is quantified rather than selected, and the local witnesses $u_x$ are existential.