Theorem 20.3.212432A5A2AE14

Proof. Write $d_ S=d_{S,\beta }$ and $a\lor b=\max (a,b)$. For potentials $u,v\in \mathrm{PSH}(Y,\beta )$ of positive mass, besides the formula for ordered potentials we use the inequality [Lemma 6.2.1, SGPT]

on a curve; here $48=3(1+1)2^{1+2}$ is the constant of that lemma in dimension one.

Put $z=\max (w_ D,w_ E)$. By lemma 20.3.20, the polar parts of the curvature measures of $w_ D$ and $w_ E$ are $D$ and $E$. By (5.27.16), the polar part of the curvature measure of $z$ is $D\wedge E$, the atomic measure with the smaller coefficient at each point. Stokes' theorem therefore gives

The formula for ordered potentials yields

Combining this equality with the two inequalities above proves both bounds, since replacing $w_ D,w_ E$ by the ordinarily equivalent $q_ D,q_ E$ does not change the singularity distance. The lower bound and the injectivity in lemma 20.3.20 make the pseudometric a metric on these envelopes. $\square$