Theorem 18.7.6EA60F3EAF248

Proof. By proposition 18.4.8, $q_ j=P_{\theta +\epsilon _ jh}[a_ j]_{\mathcal I}\sim a_ j$. Since the backgrounds decrease and $a_{j+1}\preceq a_ j$, the function $q_{j+1}$ is a competitor for $P_{\theta +\epsilon _ jh}[a_ j]$. Hence $q_{j+1}\leq q_ j$.

The inequality $a_ j\geq u$ gives $u\preceq _{\mathcal I}a_ j$. For every local divisor $E$, lemma 5.38.9 gives

By proposition 18.7.1, $q_ j\to p:=P_\theta [u]_{\mathcal I}$ in $L^1$, and $p\leq q_ j\leq 0$. The decreasing limit of the $q_ j$ is therefore bounded below by $p$; it is a $\theta $-plurisubharmonic function that agrees with $p$ almost everywhere, hence everywhere. This identifies the limit in terms of $\theta $ and the multiplier-ideal class of $u$ alone, which proves the asserted independence. $\square$