Proposition 16.25.4C32431185924

Proof. For $\varepsilon >0$ put $q=\max \{ u,v-\varepsilon \} $. Write $T_ i=\theta _ i'+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ with smooth closed real $(1,1)$-forms $\theta _ i'$ and $\varphi _ i\in \mathrm{PSH}(X,\theta _ i')$ (theorem 10.2.15). Since $q\preceq \phi $, the monotonicity of mixed masses in corollary 13.8.8, applied with $n$ factors to the potentials $q,\ldots ,q,\varphi _1,\ldots ,\varphi _{n-p}$ and $\phi ,\ldots ,\phi ,\varphi _1,\ldots ,\varphi _{n-p}$, gives $\lambda _ q(X)\leq \lambda _\phi (X)$. On the disjoint plurifine open sets $\{ u<v-\varepsilon \} $ and $\{ u>v-\varepsilon \} $, $q$ equals $v-\varepsilon $ and $u$ respectively, so plurifine locality gives

Writing $\lambda _ u(\{ u>v-\varepsilon \} )=\lambda _ u(X)-\lambda _ u(\{ u\leq v-\varepsilon \} )$, enlarging $\{ u\leq v-\varepsilon \} $ to $\{ u<v\} $ up to the common pole set of $u$ and $v$, which carries no mass, and letting $\varepsilon \downarrow 0$, we obtain the first inequality.

Now let $\int _ X\langle \theta _\phi ^ n\rangle >0$ and $u,v\in \mathcal E(X,\theta ,\phi )$. By proposition 16.22.14, $P_\theta [u]=P_\theta [v]=P_\theta [\phi ]$. With the potentials $\varphi _ i$ of the currents $T_ i$ chosen above, the preservation of mixed masses in proposition 16.21.2, applied to the $n$ potentials $u,\ldots ,u,\varphi _1,\ldots ,\varphi _{n-p}$ and to the corresponding tuples for $v$ and $\phi $, gives $\lambda _ u(X)=\lambda _ v(X)=\lambda _\phi (X)$. Thus the last two terms in the inequality cancel. Applying the inequality with $u$ and $v$ interchanged, we get $\lambda _ u(\{ v<u\} )\leq \lambda _ v(\{ v<u\} )$, and subtracting this from $\lambda _ v(X)=\lambda _ u(X)$ gives $\lambda _ v(\{ u\leq v\} )\leq \lambda _ u(\{ u\leq v\} )$. $\square$