Example 13.28.14730582377A86

Proof. Choose a compact real interval inside a coordinate disk. Construct a binary Cantor set $K$ in it: at level $k$ there are $2^ k$ pairwise disjoint intervals of length $r_ k=\exp (-4^{k+1})$, each containing its two children of level $k+1$, and $K$ is the intersection of the unions over all levels. Let $\sigma $ be the probability measure on $K$ assigning mass $2^{-k}$ to each interval of level $k$. It has no atoms. Put $u(x)=A\int _ YG(x,y)\, \mathrm{d}\sigma (y)$, so that $T_ u=A\sigma $. On the chosen compact coordinate disk, $G(x,y)\leq \log |x-y|^2+C$ for a fixed constant $C$. For $x\in K$, integrating over the interval of level $k$ containing $x$, and using $G\leq 0$ elsewhere, gives

Thus $u=-\infty $ on $K$, while $u$ is smooth off $K$, since $\mathrm{dd}^{\mathrm{c}}u=-\theta $ there. Hence $K$ is polar, $\mu _ u=A\sigma $ and $U(u)=0$. By lemma 13.28.1, $p=P_\theta [u]$ has the same polar curvature. Since this measure already has mass $A$, we get $T_ p=T_ u$, and therefore $p=u-\sup _ Yu$. The Lelong numbers of $p$ vanish because $\sigma $ has no atoms, so all its multiplier ideals are trivial and its ideal envelope is zero. Finally $\tau (p)=A$, which gives the distance by (13.28.12). $\square$