Corollary 18.39.19AF408F696ADF

Proof. In each case the reference current is $\beta +\sum _ i c_ i[D_ i]$, which is positive. Locality off the union of the divisors, together with the fact that the non-pluripolar measure does not charge this union, gives the non-pluripolar measure $\beta ^ q$ on $M$. Its mass is finite because $\beta $ is smooth and $M$ is compact, and positive by assumption. In the case of a modified base, the potential descends and its measure is $\rho _*(\beta ^ q)$, by the same descent and measure identities as before.

The multiplier ideal on the base depends only on the logarithmic divisorial orders and the bounded smooth term. The lower inclusion uses the global lower bound for the selecting potential, and the upper inclusion uses the anisotropic threshold in a polydisk where the upper bound holds; neither uses strict positivity of $\beta $. The restriction estimates concern fixed line bundles, whatever their curvature, and the Jacobian and the fixed canonical twist in the case of a modified base are unchanged. Hence the inclusions and dimension bounds in all three proofs remain valid. The reverse inclusion for the exact anisotropic model uses only its local integrability test and the local boundedness off the pole set, so it also remains valid. $\square$