Distributional Riesz measure of a plane subharmonic function

Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be a plane domain and let $u:\mathrm{\Omega }\to [-\mathrm{\infty },\mathrm{\infty })$ be subharmonic on $\mathrm{\Omega }$ (Subharmonic functions on plane domains); in particular $u$ is not identically $-\mathrm{\infty }$ on any component. Then $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ (Plane subharmonic functions are locally integrable), so for every compactly supported smooth test function $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ the Lebesgue integral ${\int }_{\mathrm{\Omega }}u\mathrm{\Delta }\phi dA$ converges absolutely: the support of $\mathrm{\Delta }\phi$ is compact and $\mathrm{\Delta }\phi$ is bounded, so ${\int }_{\mathrm{\Omega }}\mathrm{\mid }u\mathrm{\Delta }\phi \mathrm{\mid }dA\le \mathrm{\parallel }\mathrm{\Delta }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}{\int }_{\mathrm{supp}\mathrm{\Delta }\phi }\mathrm{\mid }u\mathrm{\mid }dA<\mathrm{\infty }$. The distributional Riesz functional of $u$ is

${\mu }_u(\phi ):=\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}u\mathrm{\Delta }\phi dA(\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })),$

where $\mathrm{\Delta }={\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2$ is the Laplacian, $dA$ is area Lebesgue measure, and $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is the test space of Distribution with the distributional derivative conventions of Distributional derivative. The value is complex in general and is real for real-valued test functions. The functional ${\mu }_u$ depends only on the almost-everywhere representative of $u$: if $u=v$ a.e. then $u\mathrm{\Delta }\phi =v\mathrm{\Delta }\phi$ a.e. for every test function, so the integrals agree. Linearity in $\phi$ is inherited from the linearity of differentiation and integration.

The normalization factor $(2\pi {)}^{-1}$ is chosen so that, under Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), a logarithmic point potential has unit mass at its pole: $\mathrm{\Delta }\log \mathrm{\mid }z-a\mathrm{\mid }=2\pi {\delta }_a$ in the distributional sense, that is, ${\mu }_{\log \mathrm{\mid }\cdot -a\mathrm{\mid }}={\delta }_a$ (The negative Laplacian of the fundamental solution is the unit Dirac distribution).

What is and is not asserted here. The assignment $\phi \mapsto {\mu }_u(\phi )$ is defined as a functional on test functions. That it is continuous for the test-function topology, that it is positive on nonnegative test functions, and that it is consequently integration against a unique positive Radon measure are not part of this definition; they are proved under Dependent Choice in The distributional Riesz functional of a subharmonic function is a positive Radon measure ↗, which is the well-definedness statement for the name "Riesz measure".

Sign and coefficient conventions. With the present sign convention a subharmonic function has a positive Riesz measure: for the model $u(z)=\log \mathrm{\mid }z-a\mathrm{\mid }$ under Countable Choice one has ${\mu }_u={\delta }_a$ by the normalization above, and for a compactly supported logarithmic potential $p_{\mu }=\int \log \mathrm{\mid }z-w\mathrm{\mid }d\mu (w)$ one has ${\mu }_{p_{\mu }}=\mu$ (Distributional Laplacian of a compact logarithmic potential).

Choice. Defining the functional uses no choice principle: the integral is a Lebesgue integral of an element of $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ against a fixed smooth test function. The logarithmic point-mass comparison above invokes the published fundamental-solution theorem under its stated Countable Choice hypothesis. The positive Radon measure interpretation invokes Dependent Choice for the representation and uniqueness in the well-definedness theorem.