Theorem 4.5.126736FBE7368C

Proof. We supply the uniqueness mechanism rather than assuming a simplex theorem. If a nonzero $K\mu $ generates an extreme ray of positive superharmonic functions, every restriction of $\mu $ to a neighborhood of a support point has potential proportional to $K\mu $. Normalize these restrictions to represent $K\mu $. At a finite interior test point away from the support point, positivity bounds their masses. Shrinking the neighborhoods makes their kernels converge uniformly at each other test point, so their limit is $c\delta _ y$ and $K\mu =cK_ y$. Nonproportionality of distinct kernels forces the support to be that single point. For harmonic $K\mu $ its support is on $\Delta $, so the point is in $\Delta _1$.

Choose nested compact neighborhoods $C_ j$ exhausting $\Omega $ and containing the cutoff compact in their interiors. Put $T_ j=T_{C_ j}$. For $y\in \Delta _1$, every weak cluster of the probability measures $\gamma _{y,C_ j}$ represents $K_ y$: their supports escape interior compacts, where the kernel singularities lie, and their potentials equal $K_ y$ eventually at each interior test point. Extremality and the preceding support assertion force this cluster to be $\delta _ y$. Thus $\gamma _{y,C_ j}\to \delta _ y$. Conversely, if this convergence holds along an exhaustion and $K_ y=K\mu +K\eta $, additivity and interior uniqueness give $\gamma _{y,C_ j}=\mu _{C_ j}+\eta _{C_ j}$. Weak clusters of these positive measures sum to $\delta _ y$ and represent the respective harmonic functions. Each is proportional to $\delta _ y$, proving extremality. This convergence criterion also makes $\Delta _1$ Borel: require convergence of $T_ jf_ l(y)$ to $f_ l(y)$ for a countable dense family, expressed by countable intersections and unions of the open error sets. These arguments also show that for any measure $\nu $ carried by $\Delta _1$, $\nu _{C_ j}\to \nu $ by dominated integration of $T_ jf\to f$. Consequently two such measures representing the same function agree.

For existence start with any boundary representing measure $\mu $ and take a subsequence $\mu _{C_ j}\to \nu $, still supported on $\Delta $ in the limit and representing $K\mu $. For this subsequence let $\mathcal L$ be the continuous functions $f$ on $M$ for which $T_ jf\to f$ in $\nu $-measure. It is a linear space containing constants. It is a lattice: if $g_1,g_2\in \mathcal L$, positivity gives $T_ j(g_1\wedge g_2)\le T_ jg_1\wedge T_ jg_2$. The right side tends in measure and integral to $g_1\wedge g_2$. The integral of the left side is $\int (g_1\wedge g_2)d\nu _{C_ j}= \int (g_1\wedge g_2)d\mu _{C_ j}$, tending to the same value by the chosen weak convergence. Their nonnegative difference therefore tends to zero in $L^1$, proving the lattice assertion. The space contains $C_ c(\Omega )$, whose restrictions and $T_ j$ values on $\Delta $ are eventually zero. It also contains every continuous function agreeing with $K(\cdot ,x)$ outside an interior compact: for large $j$, $T_ jf=K(\cdot ,x)=f$ on $\Delta $. These functions separate boundary points, while compact bumps separate the other points. Lattice Stone–Weierstrass makes $\mathcal L$ dense in $C(M)$. The unit contraction bound for $T_ j$ makes it uniformly closed, hence all of $C(M)$. For a countable dense family choose a common subsequence converging $\nu $-almost everywhere. At almost every $y$, $\gamma _{y,C_ j}\to \delta _ y$ along this subsequence, which is still an exhaustion; the converse criterion makes $y\in \Delta _1$. This proves existence and the mass identity follows by evaluation at $a$.

For the topology choose a countable dense family $f_ l$ in $C(M)$. The criterion just proved identifies $\Delta _1$ with

This formula requires simultaneous approximation of finitely many tests. Each inner finite intersection is relatively open; a diagonal subsequence proves its converse. Since $\Delta $ is closed in compact metric $M$, it is itself $G_\delta $, proving the assertion in $M$. $\square$