Theorem 13.14.3 (Extended non-pluripolar measure).99C7ACBB2589 Let $\theta $ be a smooth real $(1,1)$-form and $u\in \mathrm{PSH}(X,\theta )$. The measures $\mathbf1_{D_ t}(\theta +\mathrm{dd}^{\mathrm{c}}u_ t)^ n$ increase with $t$. Their limit
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13.14.4
\begin{equation} \mathrm{MA}^{\mathrm{ext}}_\theta (u) :=\lim _{t\to +\infty }\mathbf1_{\{ u>-t\} } (\theta +\mathrm{dd}^{\mathrm{c}}\max (u,-t))^ n \end{equation}
is a positive Borel measure that puts no mass on pluripolar sets; it is finite on each $D_ t$ and $\sigma $-finite on $X$. It depends only on the current $\theta +\mathrm{dd}^{\mathrm{c}}u$; in particular, it is unchanged when a constant is added to $u$. If $\Omega $ is a Hermitian form with $\theta \leq \Omega $, then
\[ \mathrm{MA}^{\mathrm{ext}}_\theta (u) \leq \mathrm{MA}^{\mathrm{ext}}_\Omega (u). \]
The total mass of $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ may be infinite.
Proof.
For $s>t$, plurifine locality on $D_ t\subset D_ s$ gives
\[ \mathbf1_{D_ t}(\theta +\mathrm{dd}^{\mathrm{c}}u_ s)^ n =\mathbf1_{D_ t}(\theta +\mathrm{dd}^{\mathrm{c}}u_ t)^ n. \]
Together with lemma 13.14.2, this proves that the measures increase. Their increasing limit is countably additive by monotone convergence, and it vanishes on pluripolar sets because each bounded product does. The comparison with $\Omega $ passes to the limit. By the compatibility just proved,
\[ \mathbf1_{D_ t}\mathrm{MA}^{\mathrm{ext}}_\theta (u) =\mathbf1_{D_ t}(\theta +\mathrm{dd}^{\mathrm{c}}u_ t)^ n, \]
and the mass of this measure is bounded by that of the bounded product $(\Omega +\mathrm{dd}^{\mathrm{c}}u_ t)^ n$. The sets $D_ t$ exhaust the complement of $\{ u=-\infty \} $, which has zero mass, so the limit is $\sigma $-finite.
For a smooth real function $f$, consider $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$ and $u'=u-f$. On a common superlevel set of $u$ and $u'$, sufficiently deep truncations of $u$ and $u'$ define the same current with bounded potentials, so their restricted products agree. These sets exhaust the complement of the common pole set, which proves invariance under this change of decomposition. By the elliptic-regularity argument in the proof of proposition 5.28.2, any two smooth decompositions of the same current differ in this way, and adding a constant is a special case.
$\square$
Pome
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