Proposition 17.3.1998BD01BE1363

Proof. All mixed measures involved have total mass $m$, so both assertions are unchanged if constants are added to $u$ and $v$; we may therefore assume $u,v\leq \phi $. Put $u_ k=\max \{ u,\phi -k\} $ and $v_ k=\max \{ v,\phi -k\} $. Fix $0\leq j\leq n$ and write $\lambda _ k=\langle \theta _{u_ k}^ j\wedge \theta _{v_ k}^{n-j}\rangle $ and $\lambda =\langle \theta _ u^ j\wedge \theta _ v^{n-j}\rangle $. We show that

On the plurifine open set $A_ k=\{ u>\phi -k\} \cap \{ v>\phi -k\} $ the measures $\lambda _ k$ and $\lambda $ agree by locality, and $u_ k-v_ k=u-v$. By lemma 17.3.18, $|u-v|$ is $\lambda $-integrable. The sets $X\setminus A_ k$ decrease to the union of the pole sets of $u$, $v$ and $\phi $, which is pluripolar and therefore $\lambda $-null, so dominated convergence gives $\int _{X\setminus A_ k}|u-v|\lambda \to 0$. It remains to show that the integral of $|u_ k-v_ k|$ against $\lambda _ k$ over $X\setminus A_ k$ tends to zero.

For $z\in \{ u,v\} $ put $D_ k(z)=\{ z\leq \phi -k\} $, so that $X\setminus A_ k=D_ k(u)\cup D_ k(v)$. We claim that

Consider $z=u$ and $j<n$, and put $q=(v_ k+\phi -k)/2$. Since $\phi -k\leq v_ k\leq \phi $, we have $q\leq \phi -k/2$ on $X$, and $u_ k=\phi -k\leq q$ on $D_ k(u)$. Moreover $\theta _{v_ k}\leq 2\theta _ q$, since $2\theta _ q-\theta _{v_ k}=\theta _\phi \geq 0$, so multilinearity gives $\lambda _ k\leq 2^{n-j}\langle \theta _{u_ k}^ j\wedge \theta _ q^{n-j}\rangle $. The potentials $q$ and $u_ k$ belong to $\mathcal E(X,\theta ,\phi )$, and the non-strict comparison principle of proposition 16.25.4, with $n-j$ factors $\theta _ q$ and the $j$ fixed factors $\theta _{u_ k}$, gives

For $j=n$ the same bound holds directly, since then $\lambda _ k=\mu _{u_ k}$ and $D_ k(u)\subset \{ u\leq \phi -k/2\} $. The measures $\mu _{u_ k}$ and $\mu _ u$ have the same total mass and agree on $\{ u>\phi -k/2\} $ by locality, so the last mass equals $\mu _ u(\{ u\leq \phi -k/2\} )$. Since exchanging $u$ and $v$ and replacing $j$ by $n-j$ does not change $\lambda _ k$, the case $z=v$ follows from the case $z=u$. Finally,

because $\phi -z$ is $\mu _ z$-integrable. This proves (17.3.21).

Since $u_ k-\phi $ and $v_ k-\phi $ take values in $[-k,0]$, we have $|u_ k-v_ k|\leq k$, and (17.3.21) shows that the integral of $|u_ k-v_ k|$ against $\lambda _ k$ over $X\setminus A_ k$ tends to zero. This proves (17.3.20). Now pass to the limit in the cocycle formula for $u_ k,v_ k$: its left side converges to $E_\theta ^\phi (v)-E_\theta ^\phi (u)$ by proposition 17.3.12, and each term on the right converges by (17.3.20). The tangent bounds pass to the limit in the same way, using the terms $j=n$ and $j=0$. Taking $v=\phi $ in the cocycle formula gives the last assertion. $\square$