Proposition 17.4.71A67C7F5BE56

Proof. Since $u\leq \max \{ u,v\} \preceq \phi $, mass monotonicity gives $\int _ X\langle \theta _{\max \{ u,v\} }^ n\rangle =m$. Thus $\int _ X\langle \theta _ u^ n\rangle +\int _ X\langle \theta _ v^ n\rangle =2m>\int _ X\langle \theta _{\max \{ u,v\} }^ n\rangle $, and proposition 16.23.1 shows that $w\neq -\infty $. Theorem 16.23.5 gives $2m\leq m+\int _ X\langle \theta _ w^ n\rangle $, while $\int _ X\langle \theta _ w^ n\rangle \leq m$ by mass monotonicity, since $w\leq u$. Hence $w\in \mathcal E(X,\theta ,\phi )$.

Now let $u,v\in \mathcal E^1(X,\theta ,\phi )$. The rooftop commutes with adding a common constant to $u$ and $v$ (proposition 16.11.4(3)), so we may assume $u,v\leq \phi $; then $w\leq \phi $. The contact inequality of corollary 16.13.5 gives

Integrate the nonnegative weight $\min \{ \phi -w,N\} $ against both sides. On the first contact set the weight equals $\min \{ \phi -u,N\} $, and on the second it equals $\min \{ \phi -v,N\} $; pole sets carry no mass. Letting $N\to \infty $ gives the displayed bound. Both terms on the right are finite, so $w\in \mathcal E^1(X,\theta ,\phi )$. $\square$