Lemma 20.3.20.A374F6B4426E The function $w_ D$ is $\beta $-plurisubharmonic, and
\[ q_ D\sim w_ D,\qquad \int _ Y\mathrm{MA}_\beta (q_ D) =\int _ Y\mathrm{MA}_\beta (w_ D)=A-D(Y). \]
The map $D\mapsto q_ D$ is a bijection from the positive atomic measures of mass at most $A$ onto the set of all envelopes $P_\beta [a]_{\mathcal I}$, $a\in \mathrm{PSH}(Y,\beta )$, and $q_ D$ has positive mass exactly when $D(Y)<A$. The measure $D$ is the atomic part of $\beta +\mathrm{dd}^{\mathrm{c}}q_ D$.
Proof.
One has
\[ \beta +\mathrm{dd}^{\mathrm{c}}w_ D=D+(A-D(Y))\eta \geq 0. \]
Every atom of $D$ is a pole of $w_ D$, and the smooth measure $\eta $ charges no polar set, including any additional poles caused by accumulation of the atoms. Hence $\mathrm{MA}_\beta (w_ D)=(A-D(Y))\eta $.
Put $q=q_ D$. By theorem 18.4.2, $q$ has the same multiplier ideals, and hence the same Lelong numbers at points, as $w_ D$. Applied to $\mathrm{dd}^{\mathrm{c}}w_ D=D-D(Y)\eta $, lemma 5.38.5 gives $q\preceq w_ D$, and the envelope inequality $q\geq w_ D-\sup _ Yw_ D$ proves $q\sim w_ D$. By corollary 5.27.15, the polar part of $\beta +\mathrm{dd}^{\mathrm{c}}q$ is therefore also $D$, and Stokes' theorem gives the non-pluripolar mass $A-D(Y)$.
For any $a\in \mathrm{PSH}(Y,\beta )$, its Lelong numbers at points give an atomic measure $D_ a$ of mass at most $A$. On a smooth curve these numbers determine all multiplier ideals, by theorem 5.37.3. Thus $a\sim _{\mathcal I}w_{D_ a}$, so $P_\beta [a]_{\mathcal I}=q_{D_ a}$. If $D_ a(Y)=A$, the atoms exhaust the whole curvature measure, and the non-pluripolar mass is zero. This proves surjectivity and the description of the potentials of positive mass. Distinct atomic measures give distinct Lelong numbers, which proves injectivity.
$\square$
Pome
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