Corollary 17.5.38.C78DFCF82004 Let $\theta $ be a smooth closed real $(1,1)$-form, and let $f\colon X\to [-\infty ,+\infty ]$ be a quasi-continuous Borel function. Suppose that the family $\mathcal A_\theta (f)$ of definition 16.11.3 is nonempty and uniformly bounded above. Then $u=P_\theta (f)$ satisfies
\[ \mu _ u\bigl (X\setminus \{ u=f\} \bigr )=0. \]
If $\{ \theta \} $ is not big, then $\mu _ u=0$. Moreover, if $r\geq 1$ and $u_1,\ldots ,u_ r\in \mathrm{PSH}(X,\theta )$ have a rooftop $w=P_\theta (\min _{1\leq j\leq r}u_ j)$ that is not identically $-\infty $, then
\[ \mu _ w\leq \sum _{j=1}^ r\mathbf1_{\{ w=u_ j\} }\mu _{u_ j}. \]
Proof.
By proposition 16.11.4(1), $u\in \mathcal A_\theta (f)$; that is, $u\in \mathrm{PSH}(X,\theta )$ and $u\leq f$ quasi-everywhere. If $\{ \theta \} $ is not big, then $\mu _ u(X)=0$ by theorem 17.5.37, since otherwise $\theta +\mathrm{dd}^{\mathrm{c}}u$ would be a closed positive current of positive non-pluripolar mass in $\{ \theta \} $. Hence $\mu _ u=0$, and in particular the first assertion holds.
If $\{ \theta \} $ is big, the first assertion is theorem 16.13.4.
For the rooftop, the function $\min _ ju_ j$ is quasi-continuous, as a finite minimum of potentials (see the beginning of section 16.11), and it is bounded above; its envelope is $w$. Since $w$ is not identically $-\infty $, the family $\mathcal A_\theta (\min _ ju_ j)$ is nonempty, and it is uniformly bounded above because the obstacle is. By the first assertion, $\mu _ w$ is carried by $\{ w=\min _ ju_ j\} \subset \bigcup _ j\{ w=u_ j\} $. The obstacle is upper semicontinuous, so $w\leq u_ j$ everywhere by proposition 16.11.4(1),(2), and lemma 16.13.1 gives $\mathbf1_{\{ w=u_ j\} }\mu _ w\leq \mathbf1_{\{ w=u_ j\} }\mu _{u_ j}$ for every $j$. Summing over $j$ proves the rooftop inequality.
$\square$
Pome
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