We return to the general setting of the chapter, with the Hermitian form $\omega $ fixed at its beginning. Locally, every finite non-pluripolar measure is absolutely continuous with respect to the Monge–Ampère measure of a bounded plurisubharmonic function (lemma 5.27.10). We globalize this statement in a prescribed ordinary type. The local potentials are glued to a strictly $\theta $-plurisubharmonic potential lying below the reference and are then truncated relative to the reference. A decreasing convex sum of the resulting countable family has a Monge–Ampère measure that dominates every local piece in the sense of absolute continuity. Its potential has the prescribed ordinary type, and the given measure has an integrable density with respect to its Monge–Ampère measure.
Proof.
Translate $\varphi $ below zero. Put $q=P_{\epsilon \theta }(\varphi )\leq \varphi $, and choose a strict big potential $\rho \leq 0$ with $\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq a\omega $, where $a>0$. Then
\[ s=q+(1-\epsilon )\rho \leq \varphi ,\qquad \theta +\mathrm{dd}^{\mathrm{c}}s\geq (1-\epsilon )a\omega . \]
In particular $s$ is $\theta $-plurisubharmonic and not identically $-\infty $. Its pole set is pluripolar, as is that of $\varphi $, so both are negligible for $\mu $.
Cover $X$ by finitely many closed coordinate balls $K_ i$, each contained in a larger coordinate ball $B_ i$. Apply lemma 5.27.10 to $\mu _ i=\mathbf1_{K_ i}\mu $. This gives bounded local plurisubharmonic $v_ i$, normalized by $1\leq v_ i\leq 2$, with
\[ \mu _ i\ll \nu _ i:=(\mathrm{dd}^{\mathrm{c}}v_ i)^ n. \]
Take a smooth cutoff $\chi _ i$ which is zero near $K_ i$ and $-3$ on a boundary collar of $B_ i$. Choose $\delta _ i>0$ so small that
\[ \theta +\mathrm{dd}^{\mathrm{c}}\bigl (s+\delta _ i(v_ i+\chi _ i)\bigr ) \geq \tfrac 12(1-\epsilon )a\omega +\delta _ i\mathrm{dd}^{\mathrm{c}}v_ i. \]
Let $S_ i$ equal $\max (s,s+\delta _ i(v_ i+\chi _ i))$ on $B_ i$ and $s$ elsewhere. The boundary collar makes this a global $\theta $-plurisubharmonic function. Since $\chi _ i\leq 0$ may be arranged,
\[ s\leq S_ i\leq s+2\delta _ i\leq \varphi +2\delta _ i. \]
For every positive integer $m$, set
\[ q_{i,m}=\max (\varphi -m,S_ i)-2\delta _ i. \]
These potentials have the same ordinary type as $\varphi $, and
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15.24.2
\begin{equation} \varphi -m-2\delta _ i\leq q_{i,m}\leq \varphi . \end{equation}
On a neighborhood of $K_ i$, $S_ i=s+\delta _ i v_ i$ away from the pole set of $s$. Let
\[ D_{i,m}=\{ s+\delta _ i v_ i>\varphi -m\} \]
in that neighborhood, restricting to points where the potentials are finite. This is a plurifinely open set there. Locality and positive mixed products give
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15.24.3
\begin{equation} \mathbf1_{D_{i,m}}\mathrm{MA}^{\mathrm{ext}}_\theta (q_{i,m}) \geq \delta _ i^ n\mathbf1_{D_{i,m}}\nu _ i. \end{equation}
Here $s$ need not be bounded. After a local smooth correction, the positive product inequality holds at every common finite truncation level of $s$ and the bounded potential $v_ i$. Exhaust the complement of the pole set of $s$. The measure $\nu _ i$ is non-pluripolar, so it loses no mass in that exhaustion. Then restrict the resulting inequality by plurifine locality. This proves (15.24.3) for the extended measures.
For $\mu _ i$-almost every point of $K_ i$, both $s$ and $\varphi $ are finite, and hence that point belongs to $D_{i,m}$ for some $m$. Consequently, a Borel set which is null for every $\mathrm{MA}^{\mathrm{ext}}_\theta (q_{i,m})$ is null for every $\mu _ i$, and therefore for $\mu $.
Enumerate the countable family $q_{i,m}$ as $q_ k$, and write its bounds as $\varphi -C_ k\leq q_ k\leq \varphi $ with $C_ k<\infty $. Choose positive $t_ k$ such that
\[ \sum _ k t_ k<1,\qquad B:=\sum _ k t_ kC_ k<\infty . \]
The finite convex combinations
\[ \psi _ N=\sum _{k=1}^ N t_ kq_ k +\left(1-\sum _{k=1}^ N t_ k\right)\varphi \]
decrease and lie between $\varphi -B$ and $\varphi $. Their limit $\psi $ is $\theta $-plurisubharmonic, satisfies $\psi \geq \varphi -B$, and hence $\psi \sim \varphi $. For any fixed $k$, separate its summand to write
\[ \psi =t_ kq_ k+(1-t_ k)w_ k, \qquad w_ k\in \mathrm{PSH}(X,\theta ),\qquad w_ k\sim \varphi . \]
The remainder is obtained by the same decreasing convex-sum construction. Positive non-pluripolar mixed products give
\[ \mathrm{MA}^{\mathrm{ext}}_\theta (\psi ) \geq t_ k^ n\mathrm{MA}^{\mathrm{ext}}_\theta (q_ k). \]
This is valid also for infinite measures: first restrict away from the common pole set, which is the pole set of $\varphi $ by the ordinary-type bounds, and then exhaust by finite truncations. All measures ignore that pole set. The preceding null-set conclusion now gives $\mu \ll \mathrm{MA}^{\mathrm{ext}}_\theta (\psi )$.
By theorem 13.14.3, the measure $\mathrm{MA}^{\mathrm{ext}}_\theta (\psi )$ is $\sigma $-finite. The Radon–Nikodym theorem applies to this $\sigma $-finite measure and the finite $\mu $. It gives a nonnegative density $f$ with
\[ \int _ X f\, \mathrm{MA}^{\mathrm{ext}}_\theta (\psi )=\mu (X)<\infty , \]
as claimed. If $\mu =0$, one may simply take $\psi =\varphi $ and $f=0$.
$\square$
Pome
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