Borel harmonicity and comparison of harmonic measure

Fix a compact nonempty $K\subseteq \mathrm{\partial }\mathrm{\Omega }$ and for $m\ge 1$ define ${\phi }_m(x)=\max \{0,1-md(x,K)\}$ on $\mathrm{\partial }\mathrm{\Omega }$. Each ${\phi }_m$ is continuous and $0\le {\phi }_{m+1}\le {\phi }_m\le 1$; and ${\phi }_m=1$ on $K$, while for $x\mathrm{\notin }K$ the compact set $K$ and the closed set $\{x\}$ are disjoint, so $d(x,K)>0$ by [F7] and ${\phi }_m(x)=0$ for all $m\ge 1\mathrm{/}d(x,K)$. Thus ${\phi }_m\downarrow {\mathbf{1}}_K$ pointwise on $\mathrm{\partial }\mathrm{\Omega }$.

Let $U\subseteq \mathrm{\partial }\mathrm{\Omega }$ be open. The cases $U=\mathrm{\partial }\mathrm{\Omega }$ and $U=\mathrm{\varnothing }$ give the constant functions $1$ and $0$, which are harmonic; assume $U\mathrm{\notin }\{\mathrm{\varnothing },\mathrm{\partial }\mathrm{\Omega }\}$. For $m\ge 1$ put $K_m=\{x\in \mathrm{\partial }\mathrm{\Omega }:d(x,\mathrm{\partial }\mathrm{\Omega }\setminus U)\ge 1\mathrm{/}m\}$. Since $\mathrm{\partial }\mathrm{\Omega }\setminus U$ is a nonempty closed set and $d(\cdot ,\mathrm{\partial }\mathrm{\Omega }\setminus U)$ is continuous, each $K_m$ is closed in $\mathrm{\partial }\mathrm{\Omega }$, hence compact because $\mathrm{\partial }\mathrm{\Omega }$ is compact; clearly $K_m\subseteq K_{m+1}$. If $x\in K_m$ then $d(x,\mathrm{\partial }\mathrm{\Omega }\setminus U)\ge 1\mathrm{/}m>0$, so $x\mathrm{\notin }\mathrm{\partial }\mathrm{\Omega }\setminus U$ (that set being closed, a point of it would have distance zero), hence $x\in U$ and $K_m\subseteq U$. Conversely every $x\in U$ lies outside the nonempty closed set $\mathrm{\partial }\mathrm{\Omega }\setminus U$, so by [F7] applied to the compact singleton $\{x\}$ we have $d(x,\mathrm{\partial }\mathrm{\Omega }\setminus U)>0$, hence $x\in K_m$ for all large $m$; therefore ${\bigcup }_m{K}_m=U$.

Now let ${\mathrm{\Omega }}_1\subseteq {\mathrm{\Omega }}_2$ be bounded regular domains, let $z\in {\mathrm{\Omega }}_1$, and let $K\subseteq \mathrm{\partial }{\mathrm{\Omega }}_1\cap \mathrm{\partial }{\mathrm{\Omega }}_2$ be compact. Such a $K$ is a compact subset of both $\mathrm{\partial }{\mathrm{\Omega }}_1$ and $\mathrm{\partial }{\mathrm{\Omega }}_2$ by [F9]. Let ${\phi }_2:\mathrm{\partial }{\mathrm{\Omega }}_2\to \mathbb{R}$ be continuous with ${\phi }_2\ge {\mathbf{1}}_K$, and let $u_2$ be the continuous harmonic extension of ${\phi }_2$ on ${\mathrm{\Omega }}_2$ provided by [F1]. Evaluating the representation of [F1] for ${\mathrm{\Omega }}_2$ at $z$ gives ${\int }_{\mathrm{\partial }{\mathrm{\Omega }}_2}{\phi }_{2}d{\omega }_{{\mathrm{\Omega }}_2}^z=u_2(z)$. Since ${\mathrm{\Omega }}_1\subseteq {\mathrm{\Omega }}_2$ we have $\overline{{\mathrm{\Omega }}_1}\subseteq \overline{{\mathrm{\Omega }}_2}$, and $u_2\ge 0$ on $\overline{{\mathrm{\Omega }}_2}$ by [F5] and ${\phi }_2\ge 0$; restricted to $\overline{{\mathrm{\Omega }}_1}$ it is continuous, harmonic on ${\mathrm{\Omega }}_1$, and hence is the unique continuous harmonic extension of its trace $u_2{\mathrm{\mid }}_{\mathrm{\partial }{\mathrm{\Omega }}_1}$. Applying the representation of [F1] to ${\mathrm{\Omega }}_1$ with that trace gives ${\int }_{\mathrm{\partial }{\mathrm{\Omega }}_1}{u}_{2}d{\omega }_{{\mathrm{\Omega }}_1}^z=u_2(z)$. Finally $u_2\ge {\mathbf{1}}_K$ at every point of $K\subseteq \mathrm{\partial }{\mathrm{\Omega }}_1\cap \mathrm{\partial }{\mathrm{\Omega }}_2$, because $u_2={\phi }_2$ on $\mathrm{\partial }{\mathrm{\Omega }}_2$, and $u_2\ge 0$ everywhere on $\mathrm{\partial }{\mathrm{\Omega }}_1$, so ${\int }_{\mathrm{\partial }{\mathrm{\Omega }}_1}{u}_{2}d{\omega }_{{\mathrm{\Omega }}_1}^z\ge {\omega }_{{\mathrm{\Omega }}_1}^z(K)$. Chaining the three displays, ${\omega }_{{\mathrm{\Omega }}_1}^z(K)\le {\int }_{\mathrm{\partial }{\mathrm{\Omega }}_2}{\phi }_{2}d{\omega }_{{\mathrm{\Omega }}_2}^z$.

The compact set $K\subseteq \mathrm{\partial }{\mathrm{\Omega }}_2$ satisfies ${\omega }_{{\mathrm{\Omega }}_2}^z(K)=\inf \{{\int }_{\mathrm{\partial }{\mathrm{\Omega }}_2}\phi d{\omega }_{{\mathrm{\Omega }}_2}^z:\phi \in C(\mathrm{\partial }{\mathrm{\Omega }}_2),\phi \ge {\mathbf{1}}_K\}$. The inequality "$\ge$" is monotonicity of the integral against the positive measure ${\omega }_{{\mathrm{\Omega }}_2}^z$; for "$\le$" fix $\epsilon >0$ and use outer regularity [F2] to choose open $V\supseteq K$ with ${\omega }_{{\mathrm{\Omega }}_2}^z(V)<{\omega }_{{\mathrm{\Omega }}_2}^z(K)+\epsilon$. If $\mathrm{\partial }{\mathrm{\Omega }}_2\setminus V=\mathrm{\varnothing }$ the constant function $1$ is admissible and $\int 1d{\omega }_{{\mathrm{\Omega }}_2}^z=1={\omega }_{{\mathrm{\Omega }}_2}^z(V)<{\omega }_{{\mathrm{\Omega }}_2}^z(K)+\epsilon$. Otherwise $\mathrm{\partial }{\mathrm{\Omega }}_2\setminus V$ is nonempty closed and disjoint from $K$, so $\delta :=d(K,\mathrm{\partial }{\mathrm{\Omega }}_2\setminus V)>0$ by [F7]; the function $\phi (x)=\max \{0,1-d(x,K)\mathrm{/}\delta \}$ is continuous by [F6], satisfies ${\mathbf{1}}_K\le \phi \le {\mathbf{1}}_V$, and hence $\int \phi d{\omega }_{{\mathrm{\Omega }}_2}^z\le {\omega }_{{\mathrm{\Omega }}_2}^z(V)<{\omega }_{{\mathrm{\Omega }}_2}^z(K)+\epsilon$. Letting $\epsilon \downarrow 0$ proves the displayed infimum.

Let $E\subseteq \mathrm{\partial }{\mathrm{\Omega }}_1\cap \mathrm{\partial }{\mathrm{\Omega }}_2$ be Borel and let $\epsilon >0$. The measure ${\omega }_{{\mathrm{\Omega }}_1}^z$ has total mass one and is outer regular on Borel sets by [F2]; applied to the Borel set $\mathrm{\partial }{\mathrm{\Omega }}_1\setminus E$ it yields an open $V\supseteq \mathrm{\partial }{\mathrm{\Omega }}_1\setminus E$ with ${\omega }_{{\mathrm{\Omega }}_1}^z(V)<{\omega }_{{\mathrm{\Omega }}_1}^z(\mathrm{\partial }{\mathrm{\Omega }}_1\setminus E)+\epsilon$. Then $C:=\mathrm{\partial }{\mathrm{\Omega }}_1\setminus V$ is closed in $\mathrm{\partial }{\mathrm{\Omega }}_1$, hence compact by [F8] and $\mathrm{\partial }{\mathrm{\Omega }}_1$ being compact, satisfies $C\subseteq E$, and ${\omega }_{{\mathrm{\Omega }}_1}^z(C)=1-{\omega }_{{\mathrm{\Omega }}_1}^z(V)>1-{\omega }_{{\mathrm{\Omega }}_1}^z(\mathrm{\partial }{\mathrm{\Omega }}_1\setminus E)-\epsilon ={\omega }_{{\mathrm{\Omega }}_1}^z(E)-\epsilon$.

For each $m$ let $u_m:=H_{{\phi }_m}$ be the corresponding envelope for $\mathrm{\Omega }$; by [F1] each $u_m$ is harmonic on $\mathrm{\Omega }$, continuous on $\bar{\mathrm{\Omega }}$, and satisfies $u_m(w)={\int }_{\mathrm{\partial }\mathrm{\Omega }}{\phi }_{m}d{\omega }_{\mathrm{\Omega }}^w$ for every $w\in \mathrm{\Omega }$. Since ${\phi }_{m+1}\le {\phi }_m$ and ${\omega }_{\mathrm{\Omega }}^w$ is a positive measure, $u_{m+1}(w)=\int {\phi }_{m+1}d{\omega }_{\mathrm{\Omega }}^w\le \int {\phi }_{m}d{\omega }_{\mathrm{\Omega }}^w=u_m(w)$ and $u_m(w)\ge 0$ for every $w$.

Combining steps 1.3 and 1.4, for every compact $K\subseteq \mathrm{\partial }{\mathrm{\Omega }}_1\cap \mathrm{\partial }{\mathrm{\Omega }}_2$ we have ${\omega }_{{\mathrm{\Omega }}_1}^z(K)\le {\inf }_{{\phi }_2\ge {\mathbf{1}}_K}{\int }_{\mathrm{\partial }{\mathrm{\Omega }}_2}{\phi }_{2}d{\omega }_{{\mathrm{\Omega }}_2}^z={\omega }_{{\mathrm{\Omega }}_2}^z(K)$; for $K=\mathrm{\varnothing }$ both sides are $0$.

Fix $w\in \mathrm{\Omega }$. The functions ${\phi }_1-{\phi }_m$ are nonnegative, measurable, and increase to ${\phi }_1-{\mathbf{1}}_K$ pointwise by step 1.1, so monotone convergence [F4] gives $\int ({\phi }_1-{\phi }_m)d{\omega }_{\mathrm{\Omega }}^w\uparrow \int ({\phi }_1-{\mathbf{1}}_K)d{\omega }_{\mathrm{\Omega }}^w$. All these integrals are finite because ${\omega }_{\mathrm{\Omega }}^w$ is a probability measure and $0\le {\phi }_1-{\phi }_m\le 1$; subtracting the common finite value $\int {\phi }_{1}d{\omega }_{\mathrm{\Omega }}^w$ gives $u_m(w)=\int {\phi }_{m}d{\omega }_{\mathrm{\Omega }}^w\downarrow {\omega }_{\mathrm{\Omega }}^w(K)$.

The sequence $-u_m$ is increasing by step 2.1 and bounded above by $0$, so the divergent alternative of [F3] is excluded and $-u_m$ converges locally uniformly on $\mathrm{\Omega }$ to a harmonic function; equivalently $u_m$ converges locally uniformly to a harmonic function $u$, and by step 3.1 $u(w)={\lim }_m{u}_m(w)={\omega }_{\mathrm{\Omega }}^w(K)$ for every $w\in \mathrm{\Omega }$. Hence $w\mapsto {\omega }_{\mathrm{\Omega }}^w(K)$ is harmonic on $\mathrm{\Omega }$ for every nonempty compact $K\subseteq \mathrm{\partial }\mathrm{\Omega }$; for $K=\mathrm{\varnothing }$ it is the constant $0$, which is harmonic.

Since every ${\omega }_{\mathrm{\Omega }}^w$ is a probability measure, $0\le {\omega }_{\mathrm{\Omega }}^w(K)\le 1$ for every $w$ and every compact $K$.

For each $w$, continuity from below for the measure ${\omega }_{\mathrm{\Omega }}^w$ gives ${\omega }_{\mathrm{\Omega }}^w(U)={\lim }_m{\omega }_{\mathrm{\Omega }}^w(K_m)$ by step 1.2. The functions $w\mapsto {\omega }_{\mathrm{\Omega }}^w(K_m)$ are harmonic by step 4.1, the sequence is increasing in $m$ and takes values in $[0,1]$ by step 5.1, so its limit $w\mapsto {\omega }_{\mathrm{\Omega }}^w(U)$ is harmonic on $\mathrm{\Omega }$ by [F3].

Let $\mathcal{A}$ be the family of Borel sets $E\subseteq \mathrm{\partial }\mathrm{\Omega }$ for which $w\mapsto {\omega }_{\mathrm{\Omega }}^w(E)$ is harmonic on $\mathrm{\Omega }$. It contains $\mathrm{\partial }\mathrm{\Omega }$ because the harmonic measure is a probability, and it contains every open set by steps 1.2 and 6.1 and the two trivial open cases there. If $A,B\in \mathcal{A}$ with $A\subseteq B$, then for every $w$ the measure identity gives ${\omega }_{\mathrm{\Omega }}^w(B\setminus A)={\omega }_{\mathrm{\Omega }}^w(B)-{\omega }_{\mathrm{\Omega }}^w(A)$; the right side is a difference of harmonic functions, hence harmonic by [F12], so $B\setminus A\in \mathcal{A}$. If $E_1\subseteq E_2\subseteq \cdots$ are in $\mathcal{A}$, then continuity from below for each measure gives ${\omega }_{\mathrm{\Omega }}^w({\bigcup }_n{E}_n)={\lim }_n{\omega }_{\mathrm{\Omega }}^w(E_n)$. This is an increasing sequence of harmonic functions bounded above by $1$, so its limit is harmonic by [F3]. Thus $\mathcal{A}$ is a lambda-system by [F11]. The open subsets of $\mathrm{\partial }\mathrm{\Omega }$ form a pi-system that generates its Borel sigma-algebra, so Dynkin's pi-lambda theorem [F11] implies that $\mathcal{A}$ contains every Borel set. Hence $w\mapsto {\omega }_{\mathrm{\Omega }}^w(E)$ is harmonic for every Borel $E$, its values lie in $[0,1]$ because each ${\omega }_{\mathrm{\Omega }}^w$ is a probability measure, and countable additivity in $E$ at fixed $w$ is the measure property of ${\omega }_{\mathrm{\Omega }}^w$. [F1, F3, F10, F11, F12, step 5.1, step 6.1, step 1.2] 8.1 The compact set $C$ of step 1.5 is a compact subset of $\mathrm{\partial }{\mathrm{\Omega }}_1\cap \mathrm{\partial }{\mathrm{\Omega }}_2$ and hence of $\mathrm{\partial }{\mathrm{\Omega }}_2$ by [F9]; if $C\mathrm{\ne }\mathrm{\varnothing }$ step 2.2 gives ${\omega }_{{\mathrm{\Omega }}_1}^z(C)\le {\omega }_{{\mathrm{\Omega }}_2}^z(C)$, while for $C=\mathrm{\varnothing }$ this inequality is trivial. Since $C\subseteq E\subseteq \mathrm{\partial }{\mathrm{\Omega }}_2$, monotonicity of ${\omega }_{{\mathrm{\Omega }}_2}^z$ gives ${\omega }_{{\mathrm{\Omega }}_2}^z(C)\le {\omega }_{{\mathrm{\Omega }}_2}^z(E)$. Therefore ${\omega }_{{\mathrm{\Omega }}_1}^z(E)<{\omega }_{{\mathrm{\Omega }}_1}^z(C)+\epsilon \le {\omega }_{{\mathrm{\Omega }}_2}^z(E)+\epsilon$ for every $\epsilon >0$, so ${\omega }_{{\mathrm{\Omega }}_1}^z(E)\le {\omega }_{{\mathrm{\Omega }}_2}^z(E)$. Dependent Choice was used only through the existence and uniqueness theorem [F1]; the compact and Borel approximation arguments and the comparison itself are choice-free.