Proposition 12.10.11.08C2A578ACE2 Let $0\leq f\in L^1$ be nonzero and $b=b_\omega (f)>0$, and write $p=f^{1/n}$. Suppose $u\in \mathrm{PSH}(X,\omega )$ satisfies $\mathrm{MA}^{\mathrm{ext}}_\omega (u)\geq bf\, \mathrm{d}V$. For every smooth strictly positive $T\in \mathscr G_\omega $, the measure
\[ R_{u,T}=(\omega +\mathrm{dd}^{\mathrm{c}}u)\wedge T-b^{1/n}p k_ T\, \mathrm{d}V \]
is positive and has mass
\[ R_{u,T}(X)=1-b^{1/n}D_ p(T). \]
In particular, if $(T_ j)$ is a sequence of smooth strictly positive members of $\mathscr G_\omega $ with $D_ p(T_ j)\to \rho _\omega (p)$, then $\| R_{u,T_ j}\| _{\mathrm{TV}}\to 0$, uniformly over all threshold subsolutions $u$.
If also $p\in \mathcal D_\omega ^0$ and $T_ j\to T$ as currents, then $T$ maximizes $D_ p$ and
\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)\wedge T_ j \rightharpoonup b^{1/n}p k_ T\, \mathrm{d}V. \]
In particular, for a fixed sequence $(T_ j)$ with limit $T$, this limiting probability is the same for all threshold subsolutions $u$.
Proof.
A smooth strictly positive $T\in \mathscr G_\omega $ is $\gamma ^{n-1}$ for a Gauduchon metric $\gamma $ normalized by $\int _ X\omega \wedge \gamma ^{n-1}=1$. Lemma 13.14.12 proves positivity of $R_{u,T}$, and Stokes' theorem for currents gives $((\omega +\mathrm{dd}^{\mathrm{c}}u)\wedge T)(X)=\int _ X\omega \wedge T=1$. This proves the mass formula. Since $\rho _\omega (p)=b^{-1/n}$, the nonnegative mass tends to zero along a maximizing sequence; it equals the total variation norm and does not depend on $u$.
Under the tail hypothesis, theorem 12.10.1 gives uniform integrability of the densities $p k_{T_ j}$. Thus proposition 12.10.6 makes $T$ a maximizer and gives $p k_{T_ j}\, \mathrm{d}V\rightharpoonup p k_ T\, \mathrm{d}V$. Adding the error measures $R_{u,T_ j}$, which tend to zero in total variation, proves the limit formula.
$\square$
Pome
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