Pome

Lemma 14.19.288068BC80F22D

Lemma 14.19.28.8068BC80F22D Let $\lambda >0$, and let $f_ a,f_ b$ be nonnegative nonzero $L^ p$ densities. Suppose that $a,b\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$ solve

\[ \omega _ a^ n=\mathrm{e}^{\lambda a}f_ a\, \mathrm{d}V,\qquad \omega _ b^ n=\mathrm{e}^{\lambda b}f_ b\, \mathrm{d}V. \]

If $f_ a\leq f_ b$ almost everywhere, then $b\leq a$ on $X$.

Proof. Since $f_ a\leq f_ b$, the two equations give $\mathrm{e}^{-\lambda b}\omega _ b^ n\geq \mathrm{e}^{-\lambda a}\omega _ a^ n$. Hence $b\leq a$ by corollary 14.19.12, applied with $u=a$ and $v=b$. $\square$

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