Theorem 10.31.32568C6F1B2F7

Theorem 10.31.3.2568C6F1B2F7 For every nonzero finite non-pluripolar measure $\mu $ on $X$:

1. There are $u\in \mathcal E_\varphi $ and a unique $c>0$ such that $\mathrm{MA}_\theta (u)=c\mu $.

2. For each $\lambda >0$ there is a unique $u\in \mathcal E_\varphi $ such that $\mathrm{MA}_\theta (u)=\mathrm{e}^{\lambda u}\mu $.

If $\mu =f\, \mathrm{d}V$ with $0\leq f\in L^ p(\mathrm{d}V)$ for some $p>1$, the solutions in both parts may be taken with $u\sim \varphi $.