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Theorem 15.26.46076D6F3256C

Theorem 15.26.4.6076D6F3256C Let $\nu $ be a finite nonzero positive measure with $M_\rho (\nu )<\infty $. Then $\nu $ is non-pluripolar and

\[ \mathrm{MA}_ h(u_\rho )\geq \frac{k}{M_\rho (\nu )}\nu . \]

For every $\lambda >0$ there is a unique bounded $U_\lambda \in \mathrm{PSH}(X,h)$ satisfying

\[ \mathrm{MA}_ h(U_\lambda )=\mathrm e^{\lambda U_\lambda }\nu , \qquad 0<\mathrm{MA}_ h(U_\lambda )(X)<\infty . \]

Proof. Since $\nu \leq M_\rho (\nu )\rho $, the reference inequality proves the first assertion. Write $\alpha =\mathrm{MA}_ h(u_\rho )$. Then $\nu =f\alpha $ with $0\leq f\leq M_\rho (\nu )/k$. The form $h>0$ is semipositive and big, so theorem 15.10.1 gives the unique bounded solution. Its boundedness and the equation give positive finite mass. $\square$

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