Pome2CA6CA22B712Lemma 15.31.3.2CA6CA22B712 For every probability $\mu $, the function $Q_\mu $ is $\kappa $-plurisubharmonic, has $\kappa $-mean zero, and satisfies
If $\mu $ is non-polar, then $\mathrm{MA}^{\mathrm{ext}}_\kappa (Q_\mu )=\mu $. Every $p\in \mathrm{PSH}(Y,\kappa )$ with $\mathrm{MA}^{\mathrm{ext}}_\kappa (p)\geq \mu $ is of the form $Q_\mu +C$.
Proof. The Poincaré–Lelong formula gives $\mathrm{dd}^{\mathrm{c}}_ xG(x,y)=\delta _ y-\kappa $. After a rotation sending $y$ to $[1:0]$, the variable $t=d_{\mathrm{ch}}(x,y)^2$ is uniformly distributed on $[0,1]$ for the measure $\kappa $. Thus
The continuous potentials $\int \max (G(x,y),-L)\, \mathrm d\mu (y)$ are $\kappa $-plurisubharmonic by lemma 5.24.1 and averaging, and decrease to $Q_\mu $. Tonelli's theorem gives its integrability and mean zero; the decreasing-limit property gives its canonical values. Distributional Fubini proves the curvature identity.
In degree one the extended measure of $p$ is
The pole set of $Q_\mu $ is polar, so non-polarity gives its asserted measure. For any $p$, distributional Stokes gives the full current $\kappa +\mathrm{dd}^{\mathrm{c}}p$ mass one, and its extended measure has mass at most one. Domination of the probability $\mu $ therefore forces both measures to equal $\mu $. The distribution $p-Q_\mu $ is harmonic on the compact connected curve, hence constant, and canonical equality gives the assertion everywhere. $\square$
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