Theorem 16.26.1.64513E83F854 Let $u,v\in \mathrm{PSH}(X,\theta )$ with $\int _ X\langle \theta _ u^ n\rangle >0$ and $P_\theta [u]\leq P_\theta [v]$, and let $a>1$ satisfy
\[ a^ n\left(\int _ X\langle \theta _ v^ n\rangle -\int _ X\langle \theta _ u^ n\rangle \right)<\int _ X\langle \theta _ v^ n\rangle . \]
Consider the obstacle $au+(1-a)v$, evaluated where $u$ and $v$ are both finite and given arbitrary Borel values on the remaining pluripolar set. Then $\eta =P_\theta (au+(1-a)v)$ is a potential of positive mass, and
\[ a^{-1}\eta +(1-a^{-1})v\leq u, \qquad P_\theta [\eta ]\leq P_\theta [u]. \]
If $P_\theta [u]=P_\theta [v]$, then every $a>1$ is allowed, $P_\theta [\eta ]=P_\theta [u]$, and $\eta $ has the same mass as $u$.
Proof.
Write $\phi =P_\theta [u]$ and $\psi =P_\theta [v]$. Then $u\preceq \phi \leq \psi $, and by proposition 16.21.2 and the monotonicity of mass,
\[ \int _ X\langle \theta _ v^ n\rangle =\int _ X\langle \theta _\psi ^ n\rangle \geq \int _ X\langle \theta _\phi ^ n\rangle =\int _ X\langle \theta _ u^ n\rangle >0. \]
Hence theorem 16.22.4, applied to $u$ and $\psi $, gives a potential $\eta '=P_\theta (au+(1-a)\psi )$ of positive mass. Choose $C$ with $v\leq \psi +C$. Since $1-a<0$, we have $\eta '-(a-1)C\leq au+(1-a)v$ quasi-everywhere, so $\eta '-(a-1)C$ is a competitor for the obstacle $au+(1-a)v$.
We show that the competitors are uniformly bounded above. Since $u$ and $v$ are finite almost everywhere, there is a Borel set $K$ of positive $\omega ^ n$-volume on which both are bounded; the obstacle is then bounded above on $K$ by a constant $D$. Let $h$ be a competitor. By proposition 10.3.1, there is a constant $B$, depending only on $X$, $\theta $ and $\omega $, such that
\[ \int _ X|h-\sup _ Xh|\, \omega ^ n\leq B. \]
Since $h\leq D$ almost everywhere on $K$, integration over $K$ gives $\sup _ Xh\leq D+B/\int _ K\omega ^ n$. Hence the envelope $\eta $ is a potential. It dominates $\eta '-(a-1)C$, so it has positive mass. The inequality $\eta \leq au+(1-a)v$, which holds quasi-everywhere, is equivalent there to $a^{-1}\eta +(1-a^{-1})v\leq u$; this inequality between quasi-plurisubharmonic functions therefore holds everywhere.
Next we compare singularity envelopes. Put $t=a^{-1}$. For all sufficiently large integers $k$, the potential $tP_\theta (\eta +k,0)+(1-t)P_\theta (v+k,0)$ is nonpositive and more singular than $t\eta +(1-t)v\leq u$; here $P_\theta (\eta +k,0)=P_\theta (\min \{ \eta +k,0\} )$, as in definition 16.25.1, and similarly for $v$. It is therefore a competitor for $\phi $. By proposition 16.25.2, $P_\theta (\eta +k,0)$ and $P_\theta (v+k,0)$ increase almost everywhere to $P_\theta [\eta ]$ and $\psi $ as $k\to \infty $. Hence
\[ tP_\theta [\eta ]+(1-t)\psi \leq \phi \]
almost everywhere, and therefore everywhere. Since $\phi \leq \psi $, subtracting $(1-t)\psi $ where $\phi $ is finite gives $P_\theta [\eta ]\leq \phi $ almost everywhere, hence everywhere.
Suppose finally that $\phi =\psi $. Then $u$ and $v$ have the same mass, so every $a>1$ is allowed, and theorem 16.22.4 shows that $\eta '$ also has this mass. Since $\eta \geq \eta '-(a-1)C$, the mass of $\eta $ is at least that of $u$, by monotonicity. Conversely, $P_\theta [\eta ]\leq \phi $ and the preservation of mass by singularity envelopes show that the mass of $\eta $ is at most that of $\phi $, which is the mass of $u$. Thus $\phi $ is nonpositive, has the same mass as $\eta $, and is less singular than $\eta $, because $\eta \preceq P_\theta [\eta ]\leq \phi $. The characterization of $P_\theta [\eta ]$ in theorem 16.22.11 therefore gives $\phi \leq P_\theta [\eta ]$, so $P_\theta [\eta ]=\phi $.
$\square$
Pome
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