25.1.3 Subharmonic functions and polar sets7A117C94C896

The characterizations of subharmonic functions by comparison with harmonic functions and by mean values, their local integrability, the Riesz decomposition and the distributional characterization are developed in [Theorem 3.2.3, Corollary 3.2.8 and Theorems 3.2.9–3.2.11, pp. 143–148, Hor07]; see theorem 3.4.5, proposition 3.4.8, theorem 3.4.14, theorem 3.4.12. Since a subharmonic function is the limit of its means at every point, the canonical representative of a distribution with positive Laplacian is determined by the distribution at every point, not only almost everywhere. Our Riesz measure is $\Delta u$, without a factor $1/(2\pi )$; thus a logarithmic pole in the plane has Riesz mass $2\pi $.

For the compactness theorem and Hartogs' lemma (theorem 3.4.19, theorem 3.4.20) compare [Theorems 3.2.12–3.2.13, pp. 149–151, Hor07]. Theorem 3.2.13 there also contains convergence statements in Sobolev spaces and against measures; the latter are discussed in the next section. Choquet's lemma and the regularized upper envelope (lemma 3.4.22, theorem 3.4.23) correspond to [Proposition 3.4.4, pp. 205–206, Hor07]; theorem 3.4.23 also asserts that the pointwise supremum of an arbitrary family is Lebesgue measurable. For the local definition of polar sets and the removable singularity theorem (definition 3.4.26, theorem 3.4.27) compare [Definition 3.4.1 and Theorem 3.4.3, pp. 203–205, Hor07]. Our proof works directly with the local subharmonic functions of the definition, and the connectedness of the complement of a closed polar set in a domain follows from the maximum principle.