Lemma 4.5.10B0A113187C30

Proof. Exhaust $F$ by compact $F_ j$. The compact reductions are potentials $G\nu _ j=K(b\nu _ j)$, with supports in $F_ j$. At a point $x$ where $u(x)$ is finite the strictly positive minimum of $K(\cdot ,x)$ on $M$ bounds their masses. Choose a weak limit $\mu $. For a sphere compactly inside $\Omega $, its average of $K(y,\cdot )$ is a finite continuous function of $y\in M$: away from the sphere use joint bounded continuity; on it use the proved continuity of spherical Green averages after cancellation of the integrable fundamental singularity. Thus weak convergence and monotone convergence of reductions identify all spherical averages of $K\mu $ and $\widehat R_ Fu$. Fatou first bounds $K\mu \le u$, so it is defined. Let the sphere radius tend to zero to get equality everywhere. For harmonic $u=h$, take $F=\Omega $ and compact neighborhoods $F_ j$ exhausting it. The reduction equals $h$ on their interiors and its measure is supported on their boundaries; these escape every interior compact, so the limit measure lies on $\Delta $.

The representing compact measure is unique by the interior Riesz identity. To prove continuity of $T_ C$, uniform continuity of $K$ on $\Delta \times C$ shows that $K_ y$ and $K_ z$ differ there by at most $\varepsilon $ when $y,z$ are close. Fix a positive harmonic kernel bounded below on $C$; its multiple $v$ majorizes one there. Order and subadditivity of reductions give $|K\gamma _{y,C}-K\gamma _{z,C}|\le \varepsilon \widehat R_ Cv$. Testing the signed bounded-density representation of $q\in C_ c^2$ above therefore bounds $|T_ Cq(y)-T_ Cq(z)|$ by a constant times $\varepsilon $. The masses are uniformly bounded since $\int K(t,a)d\gamma _{y,C}(t)=\widehat R_ CK_ y(a)\le 1$ and $\inf _{t\in C}K(t,a)>0$. Approximate restrictions of arbitrary $f\in C(M)$ uniformly on $C$ by smooth compactly supported functions; this proves continuity of $T_ Cf$. When the cutoff compact lies in $\operatorname {int}C$, the measure is on $\partial C$, where $K(t,a)=1$, and the reduction at $a$ equals $K_ y(a)=1$. Hence its mass is one, proving the last assertion. $\square$