Theorem 4.3.2253540032CE5

Proof. We use the checked Foundations polar globalization: every bounded polar set has a finite compactly supported fundamental measure whose potential is infinite on that set. For the prescribed point put $A_ j=A\cap \{ 1/j\le |x-a|\le j\} $. Apply that result and restrict the resulting measure to $\{ |y-a|>1/(2j)\} $. The removed part has a bounded kernel at each point of $A_ j$, so the restricted fundamental potential $p_ j$ is still infinite on $A_ j$ and finite at $a$. In dimension at least three choose $b_ j>0$ so that $\sum b_ j(\mu _ j(\mathbb R^ d)+p_ j(a))<\infty $. Then $\mu =\sum b_ j\mu _ j$ is finite and its positive Newtonian potential is finite at $a$ and infinite on $A=\bigcup A_ j$, by Tonelli.

In dimension two set $q_ j=p_ j-m_ j$, where $m_ j$ is the finite infimum of $p_ j$ on $\overline{B(a,j)}$. Such an infimum is finite by lsc and local integrability. Choose $b_ j>0$ with $\sum b_ jq_ j(a)<\infty $. For each fixed integer $k$, the tail $\sum _{j\ge k}b_ jq_ j$ is a monotone sum of nonnegative superharmonic functions on $B(a,k)$, finite at $a$; it is superharmonic there. Adding the finite initial sum gives an entire superharmonic function, finite at $a$ and infinite on $A$. Without a prescribed point, first choose $a$ outside the polar (hence null) set.

For the last assertion first treat $K$ contained in an interior ball $B$. Its normalized ball capacity is zero by the checked polar zero capacity. Choose decreasing compact neighborhoods $K_ j\Subset B$ with intersection $K$; their capacitary potentials $v_ j$ lie in $[0,1]$, equal one near $K$, and decrease to zero off $K$. Indeed the limit is bounded harmonic off $K$, extends harmonically across $K$ by polar removability, and is dominated by the zero-minorant potential $v_1$, so is zero. On every compact subset of $B\setminus K$ this convergence is uniform by Dini. Pass to a subsequence such that both $\operatorname {cap}_ B(K_ j)<2^{-j}$ and $v_ j<2^{-j}$ on the first $j$ members of a compact exhaustion of $B\setminus K$ (and put every support in one fixed compact subset of $B$). This is possible by decreasing capacity continuity and the preceding uniform convergence.

Write $v_ j=G_ B\mu _ j$, scaling the normalized Riesz measures by $1/c_ d$ for our kernel. Then $\sum \mu _ j(B)<\infty $. The potential of $\mu =\sum \mu _ j$ is $\sum v_ j$, infinite on $K$ and finite off $K$ in $B$. Its fundamental potential differs by a harmonic function on $B$; thus it has exactly the same finiteness set there. Off the common compact support it is finite automatically. The same distributional/harmonic-correction argument transfers it to $G_\Omega \mu $ without changing its poles. For general compact $K\Subset \Omega $, choose finitely many closed balls inside slightly larger interior balls whose interiors cover $K$; apply the construction to the polar compact intersections with these closed balls and add their measures. The sum is finite, compactly supported in $\Omega $, infinite on $K$ and finite at every point outside $K$. $\square$