Choose a resolution $\pi :Y\to X$. Two resolutions are isomorphic over the complement of a proper analytic subset of $X$, and their extended measures agree there. Since these measures do not charge pluripolar sets, the omitted analytic sets do not affect the comparison, even when the total mass is infinite.
For $u\in \mathrm{PSH}(X,\theta )$ define
96D4B9F40B0A
9.11.25
\begin{equation} \mathrm{MA}^{\mathrm{ext}}_{X,\theta }(u) =\pi _*\mathrm{MA}^{\mathrm{ext}}_{Y,\pi ^*\theta }(u\circ \pi ), \end{equation}
where $\mathrm{MA}^{\mathrm{ext}}_{Y,\pi ^*\theta }$ is the extended measure of theorem 13.14.3 on $Y$.
Theorem 9.11.26.D12D6E199E87 The right-hand side of (9.11.25) is the same for every resolution, including resolutions that modify regular points. It is a positive $\sigma $-finite Borel measure, gives zero mass to $X_{\mathrm{sing}}$ and to locally pluripolar sets, and may have infinite total mass; for bounded $u$ it is finite. Put $r_ n=\max (n-2,0)$. There are finite constants $C$ and, for each $q>r_ n$, $C_ q$, depending only on the fixed background data and, for $C_ q$, on $q$, such that for every nonpositive $u\in \mathrm{PSH}(X,\theta )$ and every $p>0$,
\[ \int _ X\mathrm{e}^{pu}\, \mathrm{MA}^{\mathrm{ext}}_{X,\theta }(u) \leq C(1+p^{-1})^{r_ n},\qquad \int _ X(1-u)^{-q}\, \mathrm{MA}^{\mathrm{ext}}_{X,\theta }(u) \leq C_ q. \]
The constants may be obtained on any one fixed resolution.
Proof.
Given two resolutions, choose a proper analytic subset $S\subset X$ containing $X_{\mathrm{sing}}$ and the images of both exceptional sets. Each map is an isomorphism over $X\setminus S$, and there the two measures are given by the same bounded local products, followed by truncation and exhaustion. On each resolution, the inverse image of $S$ is a proper analytic subset and has zero extended mass by theorem 13.14.3. Hence both pushforwards give $S$ zero mass. This proves independence, including for resolutions whose exceptional sets lie over regular points.
Positivity follows from positivity on $Y$. On $X\setminus S$ either resolution is a Borel isomorphism, so a countable exhaustion of $Y$ by sets of finite measure pushes forward to one on $X$; adjoining the null set $S$ proves $\sigma $-finiteness. A locally pluripolar subset of $X$ pulls back to a pluripolar set off the exceptional locus, and the exceptional locus itself has zero mass. This gives the asserted vanishing.
If $u$ is bounded, its pullback is bounded on the compact manifold $Y$, and local bounded-product estimates give finite total mass. For general nonpositive $u$, apply theorem 13.15.23 on $Y$ to the smooth background $\pi ^*\theta $. Pushforward and the identity $u\circ \pi $ identify the weighted integrals exactly, and the constants come from $\pi ^*\theta $ and a Hermitian metric on $Y$.
$\square$
We define convergence in capacity on $X$ as convergence in capacity of the pullbacks to a resolution, using a smooth Hermitian metric there. The local capacity comparison of section 5.29 makes the metric immaterial, and lemma 5.43.20, applied on a common resolution, makes the choice of resolution immaterial.
By theorem 13.15.23 and proper pushforward, both damped measures in the theorem depend weakly continuously on nonpositive potentials for this convergence in capacity. For nonpositive potentials $u_ j\in \mathrm{PSH}(X,\theta )$ converging in capacity, the undamped measures $\mathrm{MA}^{\mathrm{ext}}_{X,\theta }(u_ j)$ converge weakly if, in addition, their total masses are uniformly bounded and their masses on $\{ u_ j<-T\} $ tend to zero uniformly in $j$ as $T\to \infty $. Pullback preserves these mass and sublevel conditions.
General equations on singular spaces require estimates for the degeneracy of the pulled-back background and of the density; see 24.11.1. For the cone geometry below, a radial construction gives solutions in a prescribed logarithmic class and in the bounded class of a closed background.
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