Green functions exist on all bounded plane domains

Since $a$ is an interior point of the bounded domain $\mathrm{\Omega }$, the distance ${\delta }_0:=d(a,\mathrm{\partial }\mathrm{\Omega })$ is positive, and $\mathrm{\mid }z-a\mathrm{\mid }$ is bounded above on $\mathrm{\Omega }$ by the diameter of $\mathrm{\Omega }$; by [F9] the continuous function $b_a$ attains finite extrema $m:={\min }_{\mathrm{\partial }\mathrm{\Omega }}{b}_a$ and $M:={\max }_{\mathrm{\partial }\mathrm{\Omega }}{b}_a$ on $\mathrm{\partial }\mathrm{\Omega }$. Also $b_a$ is continuous: $z\mapsto \mathrm{\mid }z-a\mathrm{\mid }$ is continuous and takes values bounded away from $0$ on $\mathrm{\partial }\mathrm{\Omega }$, and $t\mapsto -\log t$ is continuous and real on positive $t$.

Every Perron lower function is dominated by $F_a$: let $v\in \mathcal{P}(b_a,\mathrm{\Omega })$ and put $w:=v-F_a$ on the bounded complex domain $\mathrm{\Omega }\setminus \{a\}$. Then $w$ is subharmonic by [F4], since $v$ is subharmonic and $F_a=-\log \mathrm{\mid }\cdot -a\mathrm{\mid }$ is harmonic on $\mathrm{\Omega }\setminus \{a\}$ by [F5]. At every boundary point of $\mathrm{\Omega }\setminus \{a\}$ the boundary limsup of $w$ is at most $0$: at $\eta \in \mathrm{\partial }\mathrm{\Omega }$ the function $F_a$ is continuous with value $b_a(\eta )$, so ${\mathrm{lim\hspace{0.17em}sup}}_{z\to \eta }w\le {\mathrm{lim\hspace{0.17em}sup}}_{z\to \eta }v-F_a(\eta )\le b_a(\eta )-b_a(\eta )=0$; at the puncture $z=a$ one has $v\le M$ on $\mathrm{\Omega }$ by [F2] while $F_a(z)\to +\mathrm{\infty }$, so ${\mathrm{lim\hspace{0.17em}sup}}_{z\to a}w\le M-\mathrm{\infty }<0$. Since $\mathrm{\Omega }\setminus \{a\}$ is a bounded complex domain and the datum $\psi \equiv 0$ is continuous on its boundary, $w\in \mathcal{P}(0,\mathrm{\Omega }\setminus \{a\})$, and [F2] applied to that domain gives $w\le 0$ on $\mathrm{\Omega }\setminus \{a\}$.

Leastness among all candidates: let $k$ be any nonnegative logarithmic-pole candidate at $a$ on $\mathrm{\Omega }$. Near $a$ the function $k+\log \mathrm{\mid }\cdot -a\mathrm{\mid }$ agrees with a harmonic function $\varphi$ on some disc $B(a,r)\subseteq \mathrm{\Omega }$ by [F1], so on $B(a,r)\setminus \{a\}$ one has $F_a-k=(F_a+\log \mathrm{\mid }\cdot -a\mathrm{\mid })-(k+\log \mathrm{\mid }\cdot -a\mathrm{\mid })=-(k+\log \mathrm{\mid }\cdot -a\mathrm{\mid })=-\varphi$, since $F_a+\log \mathrm{\mid }\cdot -a\mathrm{\mid }=0$; gluing the harmonic functions $F_a-k$ on $\mathrm{\Omega }\setminus \{a\}$ and $-\varphi$ on $B(a,r)$ along their agreement on the connected set $B(a,r)\setminus \{a\}$ produces a harmonic extension $\tilde{V}$ of $F_a-k$ to all of $\mathrm{\Omega }$.

Boundary behaviour: at a regular boundary point $\zeta \in \mathrm{\partial }\mathrm{\Omega }$ one has ${\lim }_{z\to \zeta }{h}_a(z)=b_a(\zeta )$ by [F8], while $F_a$ is continuous at $\zeta$ with $F_a(\zeta )=b_a(\zeta )$; hence ${\lim }_{z\to \zeta }{g}_{\mathrm{\Omega }}(z,a)=b_a(\zeta )-b_a(\zeta )=0$. At an irregular boundary point no limit is asserted, and none was used: the construction of $g_{\mathrm{\Omega }}$ involved only $F_a$ and the Perron envelope of $b_a$.

Let $h_a:=H_{b_a}$. By [F3] the function $h_a$ is harmonic on $\mathrm{\Omega }$, and since $m\le U_{b_a}\le M$ while $H_{b_a}$ is the limit of suprema of values of $U_{b_a}$ over shrinking discs, $m\le h_a\le M$ on $\mathrm{\Omega }$; so $h_a$ is bounded.

Consequently $U_{b_a}(z)=\sup \{v(z):v\in \mathcal{P}(b_a,\mathrm{\Omega })\}\le F_a(z)$ for every $z\in \mathrm{\Omega }\setminus \{a\}$ by [F2] and step 1.2, and then, since $F_a$ is continuous at every $z\mathrm{\ne }a$, [F3] gives $h_a(z)={\lim }_{\rho \downarrow 0}\sup \{U_{b_a}(w):\mathrm{\mid }w-z\mathrm{\mid }<\rho \}\le {\lim }_{\rho \downarrow 0}\sup \{F_a(w):\mathrm{\mid }w-z\mathrm{\mid }<\rho \}=F_a(z)\mathrm{.}$ Hence $g_{\mathrm{\Omega }}(z,a):=F_a(z)-h_a(z)\ge 0$ for $z\in \mathrm{\Omega }\setminus \{a\}$.

The extension $\tilde{V}$ of step 1.3 belongs to $\mathcal{P}(b_a,\mathrm{\Omega })$: it is harmonic, hence subharmonic, on $\mathrm{\Omega }$ by [F4], and at each $\eta \in \mathrm{\partial }\mathrm{\Omega }$ its boundary limsup is $\mathrm{lim\hspace{0.17em}sup}(F_a-k)\le b_a(\eta )-\mathrm{lim\hspace{0.17em}inf}k\le b_a(\eta )$, because $k\ge 0$. Therefore $U_{b_a}\ge \tilde{V}$ on $\mathrm{\Omega }$ by [F2] and [F3], so $h_a\ge U_{b_a}\ge \tilde{V}$; restricting to $\mathrm{\Omega }\setminus \{a\}$, where $\tilde{V}=F_a-k$, gives $F_a-h_a\le k$, that is $g_{\mathrm{\Omega }}(z,a)\le k(z)$.

The function $g_{\mathrm{\Omega }}(\cdot ,a)$ is harmonic on $\mathrm{\Omega }\setminus \{a\}$, being the difference of the harmonic functions $F_a$ and $h_a$ there by [F4] and steps 2.1, 2.2. Moreover $g_{\mathrm{\Omega }}(z,a)+\log \mathrm{\mid }z-a\mathrm{\mid }=F_a(z)+\log \mathrm{\mid }z-a\mathrm{\mid }-h_a(z)=-h_a(z)(z\in \mathrm{\Omega }\setminus \{a\}),$ and the right-hand side is harmonic on all of $\mathrm{\Omega }$ by step 2.1; so $g_{\mathrm{\Omega }}(\cdot ,a)+\log \mathrm{\mid }\cdot -a\mathrm{\mid }$ extends harmonically across $a$ and $g_{\mathrm{\Omega }}(\cdot ,a)$ is a nonnegative logarithmic-pole candidate at $a$ in the sense of [F1].

Boundedness away from the pole: fix $\delta >0$. On the set $\{z\in \mathrm{\Omega }:\mathrm{\mid }z-a\mathrm{\mid }\ge \delta \}$ the function $F_a$ satisfies $\mathrm{\mid }{F}_a(z)\mathrm{\mid }\le \max \{\mathrm{\mid }\log \delta \mathrm{\mid },\mathrm{\mid }\log R\mathrm{\mid }\}$ where $R$ is the diameter of $\mathrm{\Omega }$, and $\mathrm{\mid }{h}_a\mathrm{\mid }\le \max \{\mathrm{\mid }m\mathrm{\mid },\mathrm{\mid }M\mathrm{\mid }\}$ by step 2.1; hence $\mathrm{\mid }{g}_{\mathrm{\Omega }}(z,a)\mathrm{\mid }\le \mathrm{\mid }{F}_a(z)\mathrm{\mid }+\mathrm{\mid }{h}_a(z)\mathrm{\mid }$ is bounded there.

The candidate is strictly positive off the pole: if $g_{\mathrm{\Omega }}(w,a)=0$ for some $w\mathrm{\ne }a$, then the nonnegative harmonic function $g_{\mathrm{\Omega }}(\cdot ,a)$ on the complex domain $\mathrm{\Omega }\setminus \{a\}$ would be identically zero by [F6]; but $g_{\mathrm{\Omega }}(z,a)\ge F_a(z)-M\to +\mathrm{\infty }$ as $z\to a$ by steps 2.1 and 2.2, so $g_{\mathrm{\Omega }}(\cdot ,a)$ is unbounded and not identically zero. Hence $g_{\mathrm{\Omega }}(z,a)>0$ for every $z\in \mathrm{\Omega }\setminus \{a\}$.

Distributional normalization: on $\mathrm{\Omega }\setminus \{a\}$ one has $F_a=2\pi {\mathrm{\Phi }}_2(\cdot -a)$ by the two-dimensional branch of [F7], and $h_a\in C^2(\mathrm{\Omega })$ with $\mathrm{\Delta }{h}_a=0$; extend $g_{\mathrm{\Omega }}(\cdot ,a)$ arbitrarily at the single point $a$. By the linearity of the embedding $f\mapsto T_f$ in [F7], $T_{g_{\mathrm{\Omega }}(\cdot ,a)}=T_{F_a}-T_{h_a}=2\pi T_{{\mathrm{\Phi }}_2(\cdot -a)}-T_{h_a}$, and by the linearity of distributional differentiation and its agreement with classical differentiation on $C^2$ functions, $-\mathrm{\Delta }{T}_{g_{\mathrm{\Omega }}(\cdot ,a)}=-2\pi \mathrm{\Delta }{T}_{{\mathrm{\Phi }}_2(\cdot -a)}+\mathrm{\Delta }{T}_{h_a}=2\pi {\delta }_a-T_{\mathrm{\Delta }{h}_a}=2\pi {\delta }_a,$ because $-\mathrm{\Delta }{T}_{{\mathrm{\Phi }}_2(\cdot -a)}={\delta }_a$ by [F7] and $\mathrm{\Delta }{T}_{h_a}=T_{\mathrm{\Delta }{h}_a}=T_0=0$ by [F7] and [F4]. This is the sense in which $-{\mathrm{\Delta }}_z{g}_{\mathrm{\Omega }}(z,a)=2\pi {\delta }_a$ on $\mathrm{\Omega }$.

Since $k$ was an arbitrary nonnegative logarithmic-pole candidate, steps 3.1, 4.1 and 2.3 show that $g_{\mathrm{\Omega }}(\cdot ,a)$ is the pointwise least such candidate and is strictly positive; by [F1] it is the canonical Green kernel $g_{\mathrm{\Omega }}(\cdot ,a)$ of $\mathrm{\Omega }$ at $a$.

The stated Countable Choice is used exactly in the cited distributional identity and classical-differentiation comparison of [F7]. The cited Perron envelope theorem [F3] now uses a choice-free directed-supremum argument, and the construction of $b_a$, the comparison of Perron lower functions and the boundary limits at regular points require no additional choice principle.