Example 4.5.561A5D2A82793

Example 4.5.5.61A5D2A82793 A finite-energy potential may be infinite on a nonempty compact polar set $Z$. Choose open neighborhoods $U_ j\Supset Z$ with $C(U_ j)<2^{-4j}$ and common compact closure. Their equilibrium measures $\mu _ j$ have $G\mu _ j=1$ on $Z$ and $\| \mu _ j\| \le 2^{-2j}$. Cauchy–Schwarz and monotone convergence give finite energy for $\sum \mu _ j$, whose potential is infinite on $Z$.

The signed energy space is nevertheless incomplete. In $\mathbb R^3$ let $\sigma _ r$ be uniform mass one on the sphere of radius $r$, and put $\lambda _ j=\sigma _{1-4^{-j}}-\sigma _1$. The spherical potential formula gives $\| \lambda _ j\| ^2=(4^ j-1)^{-1}$, so the partial sums are energy Cauchy. If they had a signed Radon energy limit $\lambda $, testing against $q=G(-\Delta q/c_3)$ for $q\in C_ c^\infty $ supported in the open unit ball would show $\lambda |_{B(0,1)}=\sum _ j\sigma _{1-4^{-j}}$. Indeed only finitely many spheres meet each such support, and energy convergence passes these tests. This positive restriction has infinite mass in the closed unit ball, contradicting local finiteness of the positive variation of $\lambda $. This argument avoids passing an unbounded-mass sequence through arbitrary uniformly approximating test functions.