Lemma 17.3.18.620127F44043 Let $u\in \mathcal E(X,\theta ,\phi )$, let $v\in \mathrm{PSH}(X,\theta )$ satisfy $v\preceq \phi $, and let $\gamma \colon X\to [-\infty ,\infty ]$ be a Borel function, for instance a $\theta $-psh function, with $\gamma \geq \max \{ u,v\} $. Then, allowing infinite values,
\[ \int _ X(\gamma -u)\mu _ v \leq 2\int _ X(\gamma -u)\mu _ u +2\int _ X(\gamma -v)\mu _ v. \]
Consequently, if $u,v,w_1,\ldots ,w_ n\in \mathcal E^1(X,\theta ,\phi )$, then
\[ \int _ X|u-v|\, \langle \theta _{w_1}\wedge \cdots \wedge \theta _{w_ n}\rangle <\infty . \]
Proof.
For $t\geq 0$ we have
\[ \{ \gamma >u+2t\} \subset \{ \gamma >v+t\} \cup \{ v>u+t\} , \]
because $\gamma \leq v+t$ and $v\leq u+t$ imply $\gamma \leq u+2t$. The weights $\gamma -u$ and $\gamma -v$ are Borel functions with values in $[0,\infty ]$ on the complements of the pole sets of $u$ and $v$, respectively, and these pole sets carry no mass; the level sets above are taken on these complements. Since $u\in \mathcal E(X,\theta ,\phi )$ and $v-t\preceq \phi $, the comparison principle of proposition 16.25.4, applied with $p=n$ to $u$ and $v-t$, gives
\[ \mu _ v(\{ v>u+t\} )\leq \mu _ u(\{ v>u+t\} )+\int _ X\langle \theta _\phi ^ n\rangle -\int _ X\langle \theta _ u^ n\rangle =\mu _ u(\{ v>u+t\} ). \]
Integrating over level sets, we obtain
\[ \begin{split} \int _ X(\gamma -u)\mu _ v & \leq 2\int _0^\infty \mu _ v(\{ \gamma >v+t\} )\, \mathrm dt +2\int _0^\infty \mu _ u(\{ v>u+t\} )\, \mathrm dt\\ & \leq 2\int _ X(\gamma -v)\mu _ v+2\int _ X(\gamma -u)\mu _ u, \end{split} \]
where in the last step we used $(v-u)_+\leq \gamma -u$, which holds because $\gamma \geq \max \{ u,v\} $. All functions involved are nonnegative, so Tonelli's theorem applies without any integrability assumption.
For the second assertion, add constants to $u$ and $v$ so that $u,v\leq \phi $; this changes $|u-v|$ by at most a constant, and the mixed measure has finite mass $m$. Put $w=\frac1n\sum _ jw_ j$, which lies in $\mathcal E^1(X,\theta ,\phi )$ by corollary 17.3.17. By multilinearity, $\langle \theta _{w_1}\wedge \cdots \wedge \theta _{w_ n}\rangle \leq n^ n\mu _ w$. Choose $C$ with $w\leq \phi +C$ and apply the first assertion to $u$, $w$ and $\gamma =\phi +C$: its right side is finite because $u,w\in \mathcal E^1(X,\theta ,\phi )$. Hence $\phi -u$ is integrable for $\mu _ w$, and so for the mixed measure. The same holds for $\phi -v$, and $|u-v|\leq (\phi -u)+(\phi -v)$.
$\square$
Pome
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