Pome07B0D783C3F0Proposition 18.5.1.07B0D783C3F0 Let $u\in \mathrm{PSH}(X,\theta )$ and $f\in C^\infty (X,\mathbb R)$. Set $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$, $u'=u-f$, $m=\min _ Xf$ and $M=\max _ Xf$. Then
The same inequalities hold for the ordinary envelopes.
Proof. If $v$ is a competitor for $P_\theta [u]_{\mathcal I}$, then $v-f+m$ is $\theta '$-plurisubharmonic, nonpositive and multiplier-ideal equivalent to $u'$. Therefore $v\leq P_{\theta '}[u']_{\mathcal I}+f-m$, and taking the envelope gives the first inequality. Conversely, a competitor $v'$ in the new background gives the competitor $v'+f-M$ in the old one, which proves the second inequality. For the ordinary envelopes, the same calculation applies with the ordinary singularity order in place of multiplier-ideal equivalence. $\square$
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