Pome

Corollary 15.2.3AF890AD49AC5

Corollary 15.2.3.AF890AD49AC5 Let $u,v\in \mathrm{PSH}(X,\theta )$ satisfy $v\preceq u$, and suppose that $u$ admits a smaller-background envelope. If $\lambda >0$ and

\[ \mathrm{e}^{-\lambda u}\mathrm{MA}^{\mathrm{ext}}_\theta (u) \leq \mathrm{e}^{-\lambda v}\mathrm{MA}^{\mathrm{ext}}_\theta (v) \]

off the pole sets, then $u\geq v$. In particular, for every $\lambda >0$ and every positive measure $\mu $, the equation $\mathrm{MA}_\theta (u)=\mathrm{e}^{\lambda u}\mu $ has at most one solution in an ordinary singularity type admitting a smaller-background envelope.

Proof. Fix $t>0$. On $\{ u<v-t\} $ the assumed inequality gives $\mathrm{MA}_\theta ^{\mathrm{ext}}(u)\leq \mathrm{e}^{-\lambda t}\mathrm{MA}_\theta ^{\mathrm{ext}}(v-t)$. The theorem, applied to $u$ and $v-t$ with $c=\mathrm{e}^{-\lambda t}$, gives $u\geq v-t$; now let $t\downarrow 0$. For two solutions of the same equation in the stated type, apply this result in both directions. The weighted measures are compared off the common pole sets, which these measures do not charge, so no value of an infinite weight is needed. $\square$

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