Pome89AC64C97CA7For a positive exponent, the right-hand side $\mathrm{e}^{\lambda u}\mu $ changes when $u$ is translated, and this fixes the additive constant. At exponent zero, translation leaves the equation unchanged; instead, a positive scalar in front of the prescribed measure is needed to match its mass.
Theorem 15.37.4.C568B2958DB9 Let $\mu $ be a nonzero finite non-pluripolar measure on $X$. Then:
For every $\lambda >0$ there is a unique $u\in \mathcal E_\varphi $ with $\mu _ u=\mathrm{e}^{\lambda u}\mu $.
There are $u\in \mathcal E_\varphi $ and a unique $c>0$ with $\mu _ u=c\mu $.
The theorem is proved in subsection 15.37.30. The case of the reference zero, which has minimal singularities because $\theta \geq 0$, is treated first in theorem 15.37.28; its solutions then serve as comparison potentials for the reference $\varphi $. For general measures the theorem asserts membership in $\mathcal E_\varphi $, not a bounded difference from $\varphi $.
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