25.1.4 Energy, capacity and exceptional setsF2A7FD4CCC54

The continuity principle and the restriction of a measure to compact sets on which its potential is continuous (theorem 4.1.10) follow the argument of [Theorems 3.4.6–3.4.7, pp. 207–208, Hor07]; the approximation lemma of Lusin type used there is lemma 4.1.9. The criterion for polarity of compact sets is [Theorem 3.4.5, pp. 206–208, Hor07]. Our relative capacity of a compact set is the total mass of the Riesz measure $\Delta u$ of its regularized relative extremal function $u$, and our Green functions are positive with $-\Delta G=\delta $; the numerical values of the capacity depend on these normalizations.

Strong subadditivity, outer regularity and capacitability are developed in [Section 3.4, pp. 209–216, especially Theorems 3.4.9 and 3.4.12 and Proposition 3.4.13, Hor07]. Choquet's capacitability theorem, theorem 4.1.23, which underlies the capacitability assertion of that section, is proved here by approximating continuous images of countable intersections of countable unions of compact sets from inside by compact sets. The convergence of integrals against measures with continuous potentials in lemma 4.1.25 is part of [Theorem 3.2.13, pp. 150–151, Hor07], and the fact that the exceptional sets of the limit theorems are polar, theorem 4.1.26, corresponds to [Theorem 3.4.14, pp. 216–217, Hor07]. This polarity strengthens the equality almost everywhere of an upper envelope and its regularization in theorem 3.4.23.

For the identity between Green energy and Dirichlet energy and for quasi-continuity compare [Propositions 3.4.19 and 3.4.21, pp. 221–223, Hor07]. In proposition 4.1.29 we work with the explicit, symmetric Green function of a ball. The positivity of the Newtonian energy and of the logarithmic energy (proposition 4.1.5) is proved by expressing the kernels as integrals of Gaussians. For the logarithmic kernel, positivity and the Cauchy–Schwarz inequality are proved, and used, only for signed measures of total mass zero. The equilibrium measure is obtained by minimizing the energy, and proposition 4.1.27 describes its potential (Frostman's theorem) with our normalization of the kernels.

The example [Example 6.1.3, XiaSampleBook] uses an atom-free measure on a polar Cantor set and refers to [p. 31, Car83] for the Cantor set. Such a measure $\mu $ is constructed explicitly in proposition 4.1.12: the Cantor set $K$ is built from a rapidly decreasing sequence of lengths, so that the subharmonic function $p(z)=\int _ K\log |z-a|^2\, \mathrm d\mu (a)$ equals $-\infty $ at every point of $K$. The remaining ingredients of that example, namely the potential on the projective line, its integrability and the computation of its mass in degree one, are proposition 13.4.10, corollary 6.9.3, example 13.4.11.