Pome6BFB2EB67A74Corollary 8.13.11.6BFB2EB67A74 Fix $a>0$. If bounded admissible $u,v$ satisfy
then $u\geq v$. Consequently, for any fixed finite positive Radon measure $\nu $, the equation $H_ m(u)=e^{au}\nu $ has at most one bounded admissible solution.
Proof. For $\delta >0$, put $E_\delta =\{ u<v-\delta \} $. Multiplying the assumed measure inequality by the positive bounded Borel function $e^{au}$ gives on $E_\delta $
The last identity uses invariance under constants. Since $e^{-a\delta }<1$, 8.13.10 applied to $u,v-\delta $ gives $u\geq v-\delta $ everywhere. Letting $\delta \downarrow 0$ gives $u\geq v$. Two solutions of the stated equation have equal weighted measures $e^{-au}H_ m(u)=\nu $. Applying the order assertion in both directions proves uniqueness. $\square$
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