Proposition 16.22.147E199392108C

Proof. Write $w_ C=P_\theta (u+C,\phi )$. Each $w_ C$ with $C$ large has the singularity type of $u$, hence mass $\int _ X\langle \theta _ u^ n\rangle $, and $w_ C$ increases almost everywhere to $w$; by corollary 13.8.9, $\int _ X\langle \theta _ w^ n\rangle =\int _ X\langle \theta _ u^ n\rangle $. By corollary 16.13.5, together with $w_ C\leq \phi $ and $w_ C\leq w\leq \phi $,

As $C\to \infty $, the sets $\{ u+C\leq \phi \} $ decrease to the pole set of $u$, which is $\mu _ u$-null, because $\phi $ is bounded above; so the first term tends to zero in total variation by dominated convergence. Moreover $\mu _{w_ C}\to \mu _ w$ weakly by corollary 13.8.9. Integrating continuous functions $\chi \geq 0$ and applying lemma 16.12.1, we obtain the asserted measure bound.

If (1) holds, then $w$ and $\phi $ have the same positive mass and both belong to $\mathcal E(X,\theta ,\phi )$. By the measure bound, $\mu _ w$ gives no mass to $\{ w<\phi \} $, and the domination principle, theorem 16.22.10, gives $w\geq \phi $; hence (2). Conversely, if (2) holds, then $u$ has the mass of $w=\phi $, which is (1).

To prove that (2) implies (3), normalize $\phi \leq 0$. This is harmless: replacing $\phi $ by $\phi +c$ replaces $w_ C$ by $P_\theta (u+C-c,\phi )+c$, hence $w$ by $w+c$, so (2) is preserved, and $P_\theta [u]$ and $P_\theta [\phi ]$ do not change. Every $w_ C$ is then a nonpositive potential with the singularity type of $u$, hence a competitor for $P_\theta [u]$, so $\phi =w\leq P_\theta [u]$. Since (2) implies (1), $u$ has the positive mass of $\phi $, so $P_\theta [u]$ is a model potential by theorem 16.22.11. Monotonicity and $u\preceq \phi $ now give

Finally, if (3) holds, then by proposition 16.21.2 the potential $u$ has the mass of $P_\theta [u]=P_\theta [\phi ]$, which is the mass of $\phi $; this is (1). $\square$