Pome87B47275137287B472751372
For $u\in \mathrm{PSH}(X,\theta )$ we write $\theta _ u=\theta +\mathrm{dd}^{\mathrm{c}}u$ and denote by $\mu _ u=\langle \theta _ u^ n\rangle $ the non-pluripolar Monge–Ampère measure of $u$, as in proposition 13.9.5; its total mass $\mu _ u(X)=\int _ X\langle \theta _ u^ n\rangle $ is called the mass of $u$. When a different background is used, we say so or indicate it by a subscript. A contact set is a set $\{ u=f\} $ on which two functions agree; for Borel functions it is a Borel set, but it need not be closed (for instance, if $U\subset X$ is open, the functions $0$ and $\mathbf1_{X\setminus U}$ agree exactly on $U$).
Three properties of these measures are used throughout. They give no mass to pluripolar sets (lemma 13.2.2), in particular to pole sets, so points where a potential equals $-\infty $ can be ignored in measure-theoretic statements. They are local in the plurifine topology (proposition 13.2.6): if two potentials agree on a plurifine open set $O$, their measures agree on $O$; and for potentials $u,v$ and $c\in \mathbb R$ the set $\{ u>v+c\} $ is plurifine open by proposition 5.47.25. Finally, multilinearity and positivity give
and the mass is monotone: $\int _ X\langle \theta _ u^ n\rangle \leq \int _ X\langle \theta _ v^ n\rangle $ if $u\preceq v$, with equality if $u\sim v$, by corollary 13.8.8.
Inequalities between measures will usually be obtained from weak limits, that is, by testing against continuous functions. The following lemma justifies this step, and it records the inner regularity of finite measures on $X$.
Lemma 16.12.1.EC77D6F30D3D Let $\lambda $ and $\lambda '$ be finite positive Borel measures on $X$.
For every Borel set $E\subset X$, $\lambda (E)=\sup \{ \lambda (K):K\subset E\text{ compact}\} $.
Let $W\subset X$ be open, and suppose that $\int _ X\chi \, \mathrm d\lambda \leq \int _ X\chi \, \mathrm d\lambda '$ for every continuous function $\chi \geq 0$ with compact support in $W$. Then $\lambda (E)\leq \lambda '(E)$ for every Borel set $E\subset W$.
Proof. (1) Cover $X$ by finitely many coordinate charts $U_1,\ldots ,U_ N$, with homeomorphisms $\kappa _ i$ of $U_ i$ onto open sets $\Omega _ i\subset \mathbb R^{2n}$, and let $E\subset X$ be Borel and $\varepsilon >0$. The Borel sets $E_ i=(E\cap U_ i)\setminus (U_1\cup \cdots \cup U_{i-1})$ partition $E$. The image of the restriction of $\lambda $ to $U_ i$ under $\kappa _ i$ is a finite Borel measure on $\Omega _ i$, hence a Radon measure by proposition 2.2.2, so there is a compact set $L_ i\subset \kappa _ i(E_ i)$ whose complement in $\kappa _ i(E_ i)$ has measure less than $\varepsilon /N$. Then $K=\bigcup _ i\kappa _ i^{-1}(L_ i)$ is a compact subset of $E$ with $\lambda (E\setminus K)<\varepsilon $.
(2) Fix a metric $d$ on $X$ that induces its topology (lemma 5.2.7). Let $K\subset W$ be compact and nonempty; by compactness there is $r>0$ such that every point at distance at most $r$ from $K$ lies in $W$. For $k>1/r$ the continuous functions $\chi _ k=\max \{ 0,1-k\, d(\cdot ,K)\} $ take values in $[0,1]$, equal $1$ on $K$, have compact support in $W$, and decrease pointwise to $\mathbf1_ K$ as $k\to \infty $. By dominated convergence for the finite measures $\lambda $ and $\lambda '$,
If $E\subset W$ is Borel, then by (1), $\lambda (E)=\sup _ K\lambda (K)\leq \sup _ K\lambda '(K)\leq \lambda '(E)$, the suprema being taken over the compact sets $K\subset E$. $\square$
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