Theorem 4.4.21D7E4078B4FE9

Proof. Remove $a$ from $E$, which does not alter regularized reduction. Outside $E$ raw and regularized reductions agree. A deficit at $a$ therefore gives a defining superharmonic majorant $v$ with $v(a)<u(a)$. Lower semicontinuity of $u$ and $v\ge u$ on $E$ give the strict-gap criterion. Conversely a thinness barrier can be chosen bounded near $a$ by truncation above its value there; local exact extension followed by compact reduction gives a bounded nonnegative superharmonic $v$ on $\Omega $, with $v(a)<\alpha <\liminf _ Ev$. On a neighborhood $V$ of $a$ we have $v>\alpha $ along $E$. The peaking assumption gives $\gamma =u(a)-\sup _{\Omega \setminus V}u>0$. For sufficiently small $\lambda >0$, the function $u(a)+\lambda (v-\alpha )$ is nonnegative, majorizes $u$ on all of $E$, and is strictly below $u(a)$ at $a$. This proves the deficit.

Locally work on a ball of radius less than $1/2$. The fundamental potential $w$ of its Lebesgue measure is finite and continuous. For each $a$ in a smaller ball split off the measure of a small ball centered at $a$. Its potential $w_1$ peaks finitely at $a$: inside that ball it is a constant minus $c_ d|y-a|^2/(2d)$, and outside it is strictly radially decreasing by the spherical mean formula. The remaining potential $w_2$ is nonnegative on the original small ball. At a self-thin point of $E$,

The polar discrepancy theorem puts all these points in a polar set. Countably many smaller balls cover the domain. Polarity implies thinness by the prescribed-finite-point pole construction (remove the point itself first); the converse follows from this self-thin point assertion. $\square$