Lemma 11.41.5.CC6718D255AC The map $w\mapsto \varphi +w$ is a bijection from $\mathrm{PSH}(X,\beta )$ onto $\mathcal A_{\preceq }$, and
\[ \mathrm{MA}^{\mathrm{ext}}_\theta (\varphi +w) =\mathrm{MA}^{\mathrm{ext}}_\beta (w). \]
Its bounded members correspond to $\mathcal A_{\sim }$. The envelope $\psi $ is genuine, satisfies $\varphi \le \psi \le \varphi +K$ for some finite $K$, meets the model condition (16.28.67), and
\[ \mathcal A_{\sim }\subset \mathcal A_{\mathrm{env}} \subset \mathcal A_{\preceq }. \]
Proof.
If $u\preceq \varphi $, its generic Lelong numbers dominate the coefficients of $\Gamma $. By lemma 5.38.1, $\theta +\mathrm{dd}^{\mathrm{c}}u-[\Gamma ]\ge 0$. The locally integrable distribution $w=u-\varphi $ therefore has a canonical $\beta $-psh representative. Off $S$ it equals the difference and is bounded above. The identity $u=\varphi +w$ holds almost everywhere, hence everywhere for the canonical quasi-psh functions. Conversely, $w\in \mathrm{PSH}(X,\beta )$ has finite supremum, and
\[ \theta +\mathrm{dd}^{\mathrm{c}}(\varphi +w)=\beta +\mathrm{dd}^{\mathrm{c}}w+[\Gamma ]\ge 0, \qquad \varphi +w\preceq \varphi . \]
These maps are inverse. The bounded correspondence is theorem 11.41.1. Off $S$ the currents agree; locality of the extended products identifies their measures there. Both measures give the analytic set $S$ zero mass by theorem 13.14.3, even when $w$ has additional poles or infinite mass. This proves the measure identity on $X$ without subtracting infinite pointwise values.
For an envelope competitor $v\le 0$ with $v\preceq \varphi $, the corresponding $w=v-\varphi $ satisfies $w\le -\varphi $ off $S$. Hence
\[ \int _ Xw^+\, \mathrm dV\le \int _ X(-\varphi )\, \mathrm dV<\infty , \]
uniformly in $v$. On a fixed finite cover by inner and outer coordinate balls, add smooth quadratic corrections making $w$ locally psh. Submean at a point attaining $\sup _ Xw$, with uniform correction and volume-ratio bounds, gives $\sup _ Xw\le C_1\int _ Xw^+\, \mathrm dV+C_2\le K$. Thus $v\le \varphi +K$ canonically. The function $\varphi $ itself is a competitor, so the ordinary envelope construction gives a genuine $\theta $-psh function between $\varphi $ and $\varphi +K$. The upper bound survives upper regularization because $\varphi +K$ is upper semicontinuous. Ordinary equivalence gives $P_\theta [\psi ]=P_\theta [\varphi ]=\psi $. For $\varepsilon <1$ sufficiently close to one, $\beta -(1-\varepsilon )\theta >0$, so $\varphi $ is an $\varepsilon \theta $-psh competitor below $\psi $. Hence $P_{\varepsilon \theta }(\psi )$ is not identically $-\infty $, as required in (16.28.67).
If $u\sim \varphi \sim \psi $, choose $K_ u\ge 0$ with $\psi -K_ u\le u$. For each $t\ge 1$, the $\theta $-psh function $a_ t=\psi -tK_ u$ obeys
\[ t^{-1}a_ t+(1-t^{-1})\psi =\psi -K_ u\le u. \]
This proves relative envelope-class membership in definition 16.28.61, including the convention at $t=1$ and common poles. The other containment follows from its defining order $u\preceq \psi \sim \varphi $.
$\square$
Pome
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