Green and harmonic-measure representation with the $2\pi$ sign

$\mathrm{\Omega }$ is nonempty, bounded, open and connected, so $\bar{\mathrm{\Omega }}$ and $\mathrm{\partial }\mathrm{\Omega }$ are compact, and $u$ extends continuously to $\bar{\mathrm{\Omega }}$ with boundary values $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$. The Laplacian $\mathrm{\Delta }u$ is continuous on $\mathrm{\Omega }$, hence Borel and bounded there, and $f:=-\mathrm{\Delta }u$ lies in $L^{\mathrm{\infty }}(\mathrm{\Omega })$. Every boundary point of $\mathrm{\Omega }$ is regular, so $\mathrm{\Omega }$ is a bounded regular plane domain in the sense of [F3].

Uniform integrability of the kernel. Let $D:=\mathrm{diam}(\mathrm{\Omega })$. For every $z\in \mathrm{\Omega }$ the domain is contained in the ball $B(z,D)$, so translation invariance, polar coordinates and [F6] give ${\int }_{\mathrm{\Omega }}\mathrm{\mid }\log \mathrm{\mid }z-y\mathrm{\mid }\mathrm{\mid }dA(y)\le {\int }_{B(0,D)}\mathrm{\mid }\log \mathrm{\mid }w\mathrm{\mid }\mathrm{\mid }dA(w)=2\pi {\int }_0^{D}r\mathrm{\mid }\log r\mathrm{\mid }dr=:J_0<\mathrm{\infty },$ and this bound is uniform over $z\in \mathrm{\Omega }$.

For every pole $a\in \mathrm{\Omega }$ the canonical kernel exists. Writing $b_a:=-\log \mathrm{\mid }\cdot -a\mathrm{\mid }\in C(\mathrm{\partial }\mathrm{\Omega })$ and $h_a:=H_{b_a}$ for its regularized Perron envelope, one has $g_{\mathrm{\Omega }}(z,a)=-\log \mathrm{\mid }z-a\mathrm{\mid }-h_a(z)$ for $z\in \mathrm{\Omega }\setminus \{a\}$, the kernel is harmonic and strictly positive off $a$, its corrector $-h_a$ is harmonic on $\mathrm{\Omega }$, $g_{\mathrm{\Omega }}(z,a)\to 0$ as $z\to \zeta$ at every boundary point $\zeta$, and $-{\mathrm{\Delta }}_z{T}_{g_{\mathrm{\Omega }}(\cdot ,a)}=2\pi {\delta }_a$. In particular $\mathrm{\Omega }$ is Greenian.

A uniform logarithmic bound. Fix $c\in \mathrm{\Omega }$ and put $R:=1+{\max }_{y\in \bar{\mathrm{\Omega }}}\mathrm{\mid }y-c\mathrm{\mid }$, so that $\bar{\mathrm{\Omega }}\subset B:=D(c,R)$, where $B$ is a bounded complex domain. By monotonicity $0\le g_{\mathrm{\Omega }}(z,y)\le g_B(z,y)$ for distinct $z,y\in \mathrm{\Omega }$, and by [F1] applied on $B$, $g_B(z,y)=-\log \mathrm{\mid }z-y\mathrm{\mid }-H_y^{(B)}(z)$ with $H_y^{(B)}$ the Perron envelope of the datum $-\log \mathrm{\mid }\cdot -y\mathrm{\mid }$ on $\mathrm{\partial }B$. By [F4], $H_y^{(B)}(z)\ge {\min }_{\mathrm{\partial }B}(-\log \mathrm{\mid }\cdot -y\mathrm{\mid })=-\log (R+\mathrm{\mid }y-c\mathrm{\mid })$, so $0\le g_{\mathrm{\Omega }}(z,y)\le -\log \mathrm{\mid }z-y\mathrm{\mid }+\log (R+\mathrm{\mid }y-c\mathrm{\mid })\le \mathrm{\mid }\log \mathrm{\mid }z-y\mathrm{\mid }\mathrm{\mid }+C_{\ast },$ with $C_{\ast }:=\log (2R)$, because $R+\mathrm{\mid }y-c\mathrm{\mid }<2R$. The constant $C_{\ast }$ is independent of $z$ and $y$.

Continuity off the diagonal. For fixed $y\in \mathrm{\Omega }$ the function $z\mapsto h_y(z)=H_{b_y}(z)$ is harmonic on $\mathrm{\Omega }$ and extends continuously to $\bar{\mathrm{\Omega }}$ with boundary values $b_y$ by [F3]; hence $(z,y)\mapsto g_{\mathrm{\Omega }}(z,y)=-\log \mathrm{\mid }z-y\mathrm{\mid }-h_y(z)$ is continuous on $\{(z,y)\in \mathrm{\Omega }\times \mathrm{\Omega }:z\mathrm{\ne }y\}$ once $y\mapsto h_y$ is controlled uniformly on compact sets. Indeed, for $y,y^{\mathrm{\prime }}\in K$ with $K\subset \mathrm{\Omega }$ compact and every $z\in \mathrm{\Omega }$, $\mathrm{\mid }{h}_y(z)-h_{y^{\mathrm{\prime }}}(z)\mathrm{\mid }\le {\max }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }{b}_y-b_{y^{\mathrm{\prime }}}\mathrm{\mid }\le \frac{\mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }}{\mathrm{dist}(K,\mathrm{\partial }\mathrm{\Omega })},$ the first inequality because $\phi \mapsto H_{\phi }$ is additive and $\mathrm{\mid }{H}_{\tau }\mathrm{\mid }\le {\max }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }\tau \mathrm{\mid }$ by [F5] and [F4], and the second by [F21]. Given $(z,y)$ with $z\mathrm{\ne }y$, choose a compact $K\subset \mathrm{\Omega }$ containing $y$ and avoiding a neighbourhood of $z$; both estimates together with continuity of $z\mapsto h_y(z)$ give joint continuity at $(z,y)$. Hence the kernel is Borel measurable on that open set by [F18].

The analytic case: structure and correctors. Assume now that $\mathrm{\partial }\mathrm{\Omega }$ is real analytic in the stated sense. Fix $\zeta \in \mathrm{\partial }\mathrm{\Omega }$ with its parametrization $\gamma$ and neighbourhood $U$. Since ${\gamma }^{\mathrm{\prime }}(0)\mathrm{\ne }0$, after relabelling the two coordinates one has ${\gamma }_1^{\mathrm{\prime }}(0)\mathrm{\ne }0$, and the inverse function theorem applied to the $C^1$ map $t\mapsto {\gamma }_1(t)$ gives a $C^1$ inverse $x\mapsto t(x)$ near $x_0={\gamma }_1(0)$ with ${\gamma }_1(t(x))=x$. Hence near $\zeta$ the boundary is the graph of the $C^1$ function $x\mapsto {\gamma }_2(t(x))$ and $\mathrm{\Omega }\cap U$ is locally one of the two components of the complement of that graph; reflecting the second coordinate if necessary, a rigid change of coordinates, makes $\mathrm{\Omega }$ locally the subgraph. Therefore $\mathrm{\Omega }$ is a bounded $C^1$ domain, so $\mathrm{\partial }\mathrm{\Omega }$ carries the finite surface measure $ds$ and the continuous outward unit normal $\nu$ of [F10]. Moreover [F12] applies with $D=\mathrm{\Omega }$: every boundary point of $\mathrm{\Omega }$ is regular and the Perron corrector $h_a=-\log \mathrm{\mid }\cdot -a\mathrm{\mid }-g_{\mathrm{\Omega }}(\cdot ,a)$ is of class $C^2(\bar{\mathrm{\Omega }})$ for every $a\in \mathrm{\Omega }$. Consequently $G_{\mathrm{\Omega }}:=g_{\mathrm{\Omega }}\mathrm{/}(2\pi )$ is a Dirichlet Green function for $-\mathrm{\Delta }$ on $\mathrm{\Omega }$ whose correctors $H_a=h_a\mathrm{/}(2\pi )$ satisfy $H_a\in C^2(\bar{\mathrm{\Omega }})$, since $\mathrm{\Phi }(\xi -a)=-(2\pi {)}^{-1}\log \mathrm{\mid }\xi -a\mathrm{\mid }=H_a(\xi )$ for $\xi \in \mathrm{\partial }\mathrm{\Omega }$.

The volume potential. With $f=-\mathrm{\Delta }u\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ define $V(z):=\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}{g}_{\mathrm{\Omega }}(z,y)f(y)dA(y)(z\in \mathrm{\Omega })\mathrm{.}$ For fixed $z$ the integrand is Borel measurable in $y$ by step 3.2, and step 3.1 together with step 1.2 bounds its integral by $\frac{1}{2\pi }\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}(C_{\ast }\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+J_0)<\mathrm{\infty }$; hence $V(z)$ is a well-defined finite real number for every $z\in \mathrm{\Omega }$.

$V$ is continuous on $\mathrm{\Omega }$. Let $z_n\to z$ in $\mathrm{\Omega }$, all terms in a compact $K\subset \mathrm{\Omega }$, and let $\epsilon >0$. Put $J(\delta ):={\int }_{B(0,2\delta )}\mathrm{\mid }\log \mathrm{\mid }w\mathrm{\mid }\mathrm{\mid }dA(w)$; by [F6] the function $\mathrm{\mid }\log \mathrm{\mid }\cdot \mathrm{\mid }\mathrm{\mid }$ is integrable on $B(0,1)$, and the integrands ${\mathbf{1}}_{B(0,2\delta )}\mathrm{\mid }\log \mathrm{\mid }w\mathrm{\mid }\mathrm{\mid }$ decrease to $0$ off the origin as $\delta \downarrow 0$, so dominated convergence [F9] gives $J(\delta )\to 0$. Fix $\delta \in (0,1)$ so small that $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}(2C_{\ast }\pi {\delta }^2+2J(\delta ))<\epsilon$. On $B(z,\delta )$ the estimates of step 3.1 bound $\mathrm{\mid }{g}_{\mathrm{\Omega }}(z_n,y)-g_{\mathrm{\Omega }}(z,y)\mathrm{\mid }$ by $2C_{\ast }+\mathrm{\mid }\log \mathrm{\mid }{z}_n-y\mathrm{\mid }\mathrm{\mid }+\mathrm{\mid }\log \mathrm{\mid }z-y\mathrm{\mid }\mathrm{\mid }$, whose integral over $B(z,\delta )$ is at most $2C_{\ast }\pi {\delta }^2+2J(\delta )$ by translation invariance, so the contribution of $B(z,\delta )$ to $\mathrm{\mid }V(z_n)-V(z)\mathrm{\mid }$ is less than $\epsilon \mathrm{/}(2\pi )$. On $\mathrm{\Omega }\setminus B(z,\delta )$ one has $\mathrm{\mid }{z}_n-y\mathrm{\mid }\ge \delta \mathrm{/}2$ for large $n$, so the bound of step 3.1 gives $\mathrm{\mid }{g}_{\mathrm{\Omega }}(z_n,y)f(y)\mathrm{\mid }\le (C_{\ast }+\max \{\log D,\log (2\mathrm{/}\delta )\})\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$, an integrable majorant on the bounded set $\mathrm{\Omega }$; the integrands converge pointwise to $g_{\mathrm{\Omega }}(z,y)f(y)$ off the null set $\{z\}$ by step 3.2, so dominated convergence makes this contribution tend to $0$. Hence $V(z_n)\to V(z)$.

$V$ vanishes at the boundary. Fix $\zeta \in \mathrm{\partial }\mathrm{\Omega }$ and $\delta \in (0,1)$. For $z\in \mathrm{\Omega }\cap B(\zeta ,\delta )$, ${\int }_{\mathrm{\Omega }\cap B(\zeta ,2\delta )}{g}_{\mathrm{\Omega }}(z,y)dA(y)\le {\int }_{B(\zeta ,2\delta )}(C_{\ast }+\mathrm{\mid }\log \mathrm{\mid }z-y\mathrm{\mid }\mathrm{\mid })dA(y)\le C_{\ast }\pi (2\delta {)}^2+J(2\delta )$ by steps 3.1 and 1.2, a bound independent of $z$ that tends to $0$ with $\delta$. On the complement $\mathrm{\Omega }\setminus B(\zeta ,2\delta )$ the kernel obeys $g_{\mathrm{\Omega }}(z,y)\le C_{\ast }+\max \{\log D,\log (1\mathrm{/}\delta )\}$ for $z\in \mathrm{\Omega }\cap B(\zeta ,\delta )$, and for each fixed $y\in \mathrm{\Omega }\setminus B(\zeta ,2\delta )$ one has $g_{\mathrm{\Omega }}(z,y)\to 0$ as $z\to \zeta$ by the boundary limit of step 2.1 at the regular point $\zeta$. Given $\epsilon >0$, choose $\delta$ first and then $z$ close enough to $\zeta$; dominated convergence on the finite-measure set $\mathrm{\Omega }\setminus B(\zeta ,2\delta )$ makes the second contribution small, so ${\lim }_{z\to \zeta }V(z)=0$.

Distributional Laplacian of $V$. Let $\varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ be a real test function with compact support $K$. The double integral ${\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}\mathrm{\mid }{g}_{\mathrm{\Omega }}(z,y)f(y)\mathrm{\Delta }\varphi (z)\mathrm{\mid }dA(y)dA(z)$ is finite because for $z\in K$ the inner integral is at most $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}(C_{\ast }\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+J_0)$ by steps 3.1 and 1.2. Fubini's theorem and the definition of the distributional Laplacian therefore give $⟨\mathrm{\Delta }{T}_V,\varphi ⟩={\int }_{\mathrm{\Omega }}V\mathrm{\Delta }\varphi =\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}f(y)[{\int }_{\mathrm{\Omega }}{g}_{\mathrm{\Omega }}(z,y)\mathrm{\Delta }\varphi (z)dA(z)]dA(y)\mathrm{.}$ The inner bracket is $⟨T_{g_{\mathrm{\Omega }}(\cdot ,y)},\mathrm{\Delta }\varphi ⟩=⟨\mathrm{\Delta }{T}_{g_{\mathrm{\Omega }}(\cdot ,y)},\varphi ⟩=-2\pi \varphi (y)$ by the distributional identity of step 2.1, so $⟨\mathrm{\Delta }{T}_V,\varphi ⟩=-{\int }_{\mathrm{\Omega }}f\varphi =-⟨T_f,\varphi ⟩$: that is $\mathrm{\Delta }{T}_V=-T_f$.

The analytic case: a dense class with $C^2$ harmonic extensions. Let $A\subseteq C(\mathrm{\partial }\mathrm{\Omega },\mathbb{R})$ be the set of restrictions to $\mathrm{\partial }\mathrm{\Omega }$ of polynomials in the two real coordinates. Then $A$ contains the constants, is closed under sums and products, and separates points of the compact metric space $\mathrm{\partial }\mathrm{\Omega }$; by [F15] it is uniformly dense in $C(\mathrm{\partial }\mathrm{\Omega },\mathbb{R})$. Fix $\varphi =p{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}\in A$. Complexifying the power series of $\gamma$ at $0$ gives a holomorphic $\mathrm{\Gamma }$ on a disc $D(0,r)$ with $\mathrm{\Gamma }(0)=\zeta$, $\mathrm{\Gamma }=\gamma$ on the real interval and ${\mathrm{\Gamma }}^{\mathrm{\prime }}(0)\mathrm{\ne }0$; by the holomorphic inverse function theorem, after shrinking $r$, $\mathrm{\Gamma }$ is a biholomorphism onto a neighbourhood $V$ of $\zeta$ that maps the upper half-disc onto $\mathrm{\Omega }\cap V$ (replacing $\gamma$ by $t\mapsto \gamma (-t)$ if necessary). Put $w:=H_{\varphi }\circ \mathrm{\Gamma }$; by [F14] and [F3] the function $w$ is harmonic on the half-disc and continuous on its closure with $w(t)=\varphi (\gamma (t))$ for $t\in (-r,r)$. Here $t\mapsto \varphi (\gamma (t))=p(\gamma (t))$ is real analytic by [F13], so it equals ${\sum }_n{a}_n{t}^n$ on some interval $(-r^{\mathrm{\prime }},r^{\mathrm{\prime }})$; the sum $F(w):={\sum }_n{a}_n{w}^n$ is holomorphic on $\mathrm{\mid }w\mathrm{\mid }<r^{\mathrm{\prime }}$, both components of $F$ are smooth by [F13], so the real part $u_{\mathrm{l}\mathrm{o}\mathrm{c}}:=\mathrm{Re}F$ is harmonic there by the $C^2$ components theorem with $u_{\mathrm{l}\mathrm{o}\mathrm{c}}(t)=\varphi (\gamma (t))$ on the edge, and $v:=w-u_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is harmonic on the half-disc, continuous on its closure and zero on the edge. By [F14] the odd reflection of (a rescaled) $v$ is harmonic on the full disc, so $v$ is $C^2$ up to the edge and $w=u_{\mathrm{l}\mathrm{o}\mathrm{c}}+v$ is $C^2$ on the closed half-disc; transferring through the biholomorphism $\mathrm{\Gamma }$ shows that $H_{\varphi }$ agrees near the arc with a $C^2$ function on a neighbourhood of the boundary. As $\zeta$ was arbitrary, $H_{\varphi }\in C^2(\bar{\mathrm{\Omega }})$, and $H_{\varphi }$ is harmonic with $\mathrm{\Delta }{H}_{\varphi }=0$.

The kernel in terms of the Green function. By symmetry [F2], $g_{\mathrm{\Omega }}(x,y)=g_{\mathrm{\Omega }}(y,x)$ for distinct $x,y$, and by step 3.3 the function $y\mapsto g_{\mathrm{\Omega }}(y,x)=-\log \mathrm{\mid }y-x\mathrm{\mid }-h_x(y)$ is of class $C^2$ near $\mathrm{\partial }\mathrm{\Omega }$; hence the trace $y\mapsto g_{\mathrm{\Omega }}(x,y)$ has a classical normal derivative ${\mathrm{\partial }}_{{\nu }_y}{g}_{\mathrm{\Omega }}(x,y)$ there. The boundary-slot derivative of [F11] is the trace of $D_z(\mathrm{\Phi }(z-x)-H_x(z))\cdot \nu (y)$, and $\mathrm{\Phi }(z-x)-H_x(z)=g_{\mathrm{\Omega }}(z,x)\mathrm{/}(2\pi )$; by symmetry, ${\mathrm{\partial }}_{{\nu }_y}{g}_{\mathrm{\Omega }}(z,x){\mathrm{\mid }}_{z=y}={\mathrm{\partial }}_{{\nu }_y}{g}_{\mathrm{\Omega }}(x,y)$. Therefore $P_{\mathrm{\Omega }}(x,y)=-\frac{1}{2\pi }{\mathrm{\partial }}_{{\nu }_y}{g}_{\mathrm{\Omega }}(x,y)$ for every $y\in \mathrm{\partial }\mathrm{\Omega }$.

$u-V$ is distributionally harmonic. Since $u\in C^2(\mathrm{\Omega })$, [F7] gives $\mathrm{\Delta }{T}_u=T_{\mathrm{\Delta }u}=T_{-f}$; subtracting the identity of step 4.4 and using linearity of the embedding and of distributional differentiation, $\mathrm{\Delta }{T}_{u-V}=T_{-f}+T_f=0$ as distributions on $\mathrm{\Omega }$.

Applying the PDE representation formula. In the analytic case, steps 3.3 and 4.5 provide: the bounded $C^1$ domain $\mathrm{\Omega }$; the Dirichlet Green function $G_{\mathrm{\Omega }}=g_{\mathrm{\Omega }}\mathrm{/}(2\pi )$ with $C^2(\bar{\mathrm{\Omega }})$ correctors; and the uniformly dense class $A$ of continuous data each of which has a harmonic $C^2(\bar{\mathrm{\Omega }})$ extension, namely $H_{\varphi }$. In the alternative hypothesis of the statement the corresponding dense class and $C^2$ harmonic extensions, together with the $C^2$ corrector condition, are assumed, and the assumed extension of $\varphi$ coincides with $H_{\varphi }$ by the uniqueness in [F3]. In both cases [F11] applies with $U:=H_{\varphi }$ for $\varphi \in A$, and $\mathrm{\Delta }{H}_{\varphi }=0$, so $H_{\varphi }(x)={\int }_{\mathrm{\partial }\mathrm{\Omega }}{P}_{\mathrm{\Omega }}(x,y)\varphi (y)dS(y),P_{\mathrm{\Omega }}(x,y)=-{\mathrm{\partial }}_{{\nu }_y}{G}_{\mathrm{\Omega }}(x,y)\ge 0,{\int }_{\mathrm{\partial }\mathrm{\Omega }}{P}_{\mathrm{\Omega }}(x,y)dS(y)=1$ for every $x\in \mathrm{\Omega }$; and by [F3], ${\int }_{\mathrm{\partial }\mathrm{\Omega }}\varphi d{\omega }_{\mathrm{\Omega }}^x=H_{\varphi }(x)$.

$u-V$ is harmonic. By [A1] Countable Choice holds, so [F8] applies and there is a unique smooth harmonic $h$ on $\mathrm{\Omega }$ with $T_{u-V}=T_h$; injectivity of the embedding modulo almost-everywhere equality gives $u-V=h$ almost everywhere. Both $u-V$ (by step 4.2 and continuity of $u$) and $h$ are continuous on $\mathrm{\Omega }$, so the set where they differ is open and null, hence empty: a nonempty open set contains a ball of radius $r>0$, whose area is $\pi r^2>0$ by the polar and translation formulas in [F6]; therefore $u-V=h$ everywhere on $\mathrm{\Omega }$.

The boundary values and harmonic measure. By step 4.3, $V(z)\to 0$ as $z\to \zeta$ for every $\zeta \in \mathrm{\partial }\mathrm{\Omega }$, while $u(z)\to u(\zeta )$ by continuity; hence $h(z)=u(z)-V(z)\to u(\zeta )$. Thus $h$ extends continuously to $\bar{\mathrm{\Omega }}$ with boundary values $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$, and the uniqueness of the continuous harmonic extension in [F3] gives $h(z)=H_{u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}}(z)={\int }_{\mathrm{\partial }\mathrm{\Omega }}ud{\omega }_{\mathrm{\Omega }}^z$ for every $z\in \mathrm{\Omega }$.

The representation formula. For $z\in \mathrm{\Omega }$, $u(z)=h(z)+V(z)={\int }_{\mathrm{\partial }\mathrm{\Omega }}ud{\omega }_{\mathrm{\Omega }}^z+\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}{g}_{\mathrm{\Omega }}(z,y)f(y)dA(y)={\int }_{\mathrm{\partial }\mathrm{\Omega }}ud{\omega }_{\mathrm{\Omega }}^z-\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}{g}_{\mathrm{\Omega }}(z,y)\mathrm{\Delta }u(y)dA(y)\mathrm{.}$ The boundary integral is absolutely finite because ${\omega }_{\mathrm{\Omega }}^z$ is a probability measure, and the volume integral because $\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}\mathrm{\mid }{g}_{\mathrm{\Omega }}(z,y)\mathrm{\Delta }u(y)\mathrm{\mid }dA(y)\le \frac{1}{2\pi }\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{\mathrm{\infty }}(C_{\ast }\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+J_0)$ by steps 3.1 and 1.2. Dependent Choice supplies harmonic measure through [F3] and implies the Countable Choice used in the kernel's distributional normalization [F1], Weyl's lemma [F8], and the measure and integration interfaces [F6], [F9], [F19] and [F20]; the density clause also uses it through [F11], [F12] and [F16]. No stronger choice principle is used. The statement is formulated for real $u$; a complex-valued $u$ is handled by applying the result to its real and imaginary parts. This proves clause 1.

Passage to all continuous data and identification of the measure. For $\varphi \in A$ and $x\in \mathrm{\Omega }$, steps 5.2 and 4.6 give ${\int }_{\mathrm{\partial }\mathrm{\Omega }}\varphi d{\omega }_{\mathrm{\Omega }}^x={\int }_{\mathrm{\partial }\mathrm{\Omega }}\varphi (y)P_{\mathrm{\Omega }}(x,y)dS(y)$. Define $\nu (E):={\int }_E{P}_{\mathrm{\Omega }}(x,y)dS(y)$ for Borel $E\subseteq \mathrm{\partial }\mathrm{\Omega }$, the surface integral of the nonnegative Borel function $P_{\mathrm{\Omega }}(x,\cdot ){\mathbf{1}}_E$ as in [F10]. Countable additivity of $\nu$ follows from the finite chart sum defining $dS$ and additivity of the Lebesgue integral over countable families of nonnegative functions [F19]; and $\nu (\mathrm{\partial }\mathrm{\Omega })={\int }_{\mathrm{\partial }\mathrm{\Omega }}{P}_{\mathrm{\Omega }}(x,y)dS(y)=1$ by step 5.2. The boundary $\mathrm{\partial }\mathrm{\Omega }$ is a compact metric subspace of ${\mathbb{R}}^2$, and the intersections with $\mathrm{\partial }\mathrm{\Omega }$ of the rational open boxes of ${\mathbb{R}}^2$ form a countable basis of its topology, so $\mathrm{\partial }\mathrm{\Omega }$ is a second-countable locally compact Hausdorff space and [F16] makes $\nu$ Radon. For $\varphi \in A$ the two probability integrals agree, and if $\psi \in C(\mathrm{\partial }\mathrm{\Omega },\mathbb{R})$ is arbitrary then uniform density of $A$ and the bound $\mathrm{\parallel }\psi -\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}$ for both probability measures extend the identity to $\psi$; hence ${\int }_{\mathrm{\partial }\mathrm{\Omega }}\psi d\nu =H_{\psi }(x)$ for every continuous $\psi$, that is, $\nu$ is a harmonic measure for $\mathrm{\Omega }$ at $x$. By the uniqueness in [F3], $\nu ={\omega }_{\mathrm{\Omega }}^x$, and step 4.6 converts this into ${\omega }_{\mathrm{\Omega }}^x(E)=-\frac{1}{2\pi }{\int }_E{\mathrm{\partial }}_{{\nu }_y}{g}_{\mathrm{\Omega }}(x,y)ds(y)$ for every Borel set $E\subseteq \mathrm{\partial }\mathrm{\Omega }$, which is the density clause. In the analytic case this used [F12] and the polynomial class; in the alternative case it used the assumed dense class and correctors. No pointwise Poisson density for arbitrary continuous data and no $C^2(\mathrm{\Omega })\cap C^1(\bar{\mathrm{\Omega }})$ representation without the bounded-Laplacian hypothesis is claimed. ∎