16.11 Obstacles and rooftopsFBD81D436A6A

Proof. Assertions (1) and (3) are contained in lemma 13.7.1: (1) is its inequality for Borel sets, applied with the roles of $D$ and $D'$ exchanged, and (3) is its final assertion.

For (2) we first observe that if $W\subset D$ is open, then every competitor $v$ for $\operatorname {Cap}_ D$ restricts to a plurisubharmonic function on $W$ with values in $[0,1]$, and the restriction of the Bedford–Taylor measure $(\mathrm{dd}^{\mathrm{c}}v)^ n$ to $W$ is the Bedford–Taylor measure of this restriction. Hence $\operatorname {Cap}_ D(E)\leq \operatorname {Cap}_ W(E)$ for every Borel set $E\subset W$, and, taking infima over the open sets $G$ with $A\subset G\subset W$, also $\operatorname {Cap}_ D^*(A)\leq \operatorname {Cap}_ W^*(A)$ for every set $A\subset W$.

Now cover $D$ by countably many open balls $B_ k$, in the coordinates of $D$, with $\overline{B_ k}\subset D$. Since the condition in definition 5.47.2 is local, $E\cap B_ k$ is pluripolar in $B_ k$, so $\operatorname {Cap}_{B_ k}^*(E\cap B_ k)=0$ by theorem 5.47.8. By the observation above, $\operatorname {Cap}_ D^*(E\cap B_ k)=0$, and the countable subadditivity of $\operatorname {Cap}_ D^*$ (proposition 7.4.3) gives $\operatorname {Cap}_ D^*(E\cap D)=0$. $\square$

Proof. (1) By lemma 5.4.2 every point of $X$ has a neighborhood on which $\theta '=\mathrm{dd}^{\mathrm{c}}h$ for a smooth function $h$, so $X$ is covered by finitely many coordinate balls $D_ i$ on which $\theta '=\mathrm{dd}^{\mathrm{c}}h_ i$ with $h_ i$ smooth near $\overline{D_ i}$. On $D_ i$ the function $u+h_ i$ is plurisubharmonic by lemma 6.14.4(2), hence quasi-continuous by theorem 7.5.2, and so is $u=(u+h_ i)-h_ i$, because $h_ i$ is continuous and real. By lemma 16.11.1(3), $u$ is quasi-continuous on every chart.

(2) Let $f_1,\ldots ,f_ k$ be quasi-continuous, let $K$ be a compact subset of a chart $D$, and let $\varepsilon >0$. Choose open sets $G_ i\subset D$ with $\operatorname {Cap}_ D(G_ i)<\varepsilon /k$ such that $f_ i$ is continuous on $K\setminus G_ i$. By subadditivity, $G=G_1\cup \cdots \cup G_ k$ has capacity less than $\varepsilon $, and all the $f_ i$, hence also their maximum and minimum, are continuous on $K\setminus G$.

(3) Let $D$ be a chart, $K\subset D$ compact and $\varepsilon >0$. The set $\{ f\ne g\} $ is pluripolar, so by lemma 16.11.1(2) its intersection with $D$ is contained in an open set $G'\subset D$ with $\operatorname {Cap}_ D(G')<\varepsilon /2$. If $G\subset D$ is an open set with $\operatorname {Cap}_ D(G)<\varepsilon /2$ such that $f$ is continuous on $K\setminus G$, then $\operatorname {Cap}_ D(G\cup G')<\varepsilon $, and $g=f$ is continuous on $K\setminus (G\cup G')$. $\square$

Proof. (1) Cover $X$ by finitely many coordinate balls on which $\theta $ has smooth local potentials $h$ (lemma 5.4.2). Applying theorem 5.10.2 on each ball to the plurisubharmonic functions $v+h$, $v\in \mathcal A_\theta (f)$, and taking the union of the resulting countable subfamilies, we find competitors $v_1,v_2,\ldots $ with $(\sup _ kv_ k)^*=P_\theta (f)$. The finite maxima $w_ k=\max \{ v_1,\ldots ,v_ k\} $ are again competitors, and they increase with $k$. Each $w_ k$ satisfies $w_ k\leq f$ outside a pluripolar set $N_ k$, and $N=\bigcup _ kN_ k$ is pluripolar by theorem 5.47.8. By corollary 5.47.11, $\sup _ kw_ k=P_\theta (f)$ outside a pluripolar set $N'$. Hence $P_\theta (f)\leq f$ outside $N\cup N'$. Since the family is nonempty and uniformly bounded above, $P_\theta (f)$ is $\theta $-plurisubharmonic by theorem 5.10.2. Thus $P_\theta (f)$ is a competitor, and by construction it dominates every competitor.

(2) Let $v$ be a competitor, let $x\in X$, and choose a coordinate ball $B(x,r)$ and a smooth function $h$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $ near its closure. Since $v+h$ is plurisubharmonic and $v\leq f$ outside a Lebesgue null set, the sub-mean value inequality gives

As $r\to 0$, the right side tends to at most $f(x)+h(x)$, because $f+h$ is upper semicontinuous. Hence $v(x)\leq f(x)$.

(3) A modification of $f$ on a pluripolar set does not change $\mathcal A_\theta (f)$, since the union of two pluripolar sets is pluripolar. Moreover $\mathcal A_\theta (f+c)=\mathcal A_\theta (f)+c$, and $\mathcal A_\theta (f)\subset \mathcal A_\theta (g)$ if $f\leq g$ quasi-everywhere. These relations between the families of competitors give the three assertions.

(4) By (3), $\mathcal A_\theta (f)-c\subset \mathcal A_\theta (g)\subset \mathcal A_\theta (f)+c$. Hence $\mathcal A_\theta (g)$ is nonempty and uniformly bounded above, and the two inequalities follow from (3).

(5) By (3) the envelopes $P_\theta (f_ j)$ decrease; if some family $\mathcal A_\theta (f_ j)$ is empty, then so are all later ones and $\mathcal A_\theta (f)$. Let $v$ be the pointwise limit of $P_\theta (f_ j)$. Suppose first that $v(x_0)>-\infty $ for some $x_0$. Then $\sup _ XP_\theta (f_ j)\geq v(x_0)$ for every $j$, so $v\in \mathrm{PSH}(X,\theta )$ by corollary 10.3.6. By (1), $v\leq P_\theta (f_ j)\leq f_ j$ outside a pluripolar set $N_ j$, so $v\leq \inf _ jf_ j=f$ outside the pluripolar set $\bigcup _ jN_ j$. Thus $v$ is a competitor for $f$ and $v\leq P_\theta (f)$; the reverse inequality follows from $P_\theta (f)\leq P_\theta (f_ j)$. Suppose next that $v\equiv -\infty $. A competitor $w$ for $f$ would be a competitor for every $f_ j$, so $w\leq P_\theta (f_ j)$ for all $j$ and $w\leq v$, which is impossible. Hence $\mathcal A_\theta (f)=\varnothing $ and $P_\theta (f)=-\infty =v$. $\square$

Proof. Commutativity, monotonicity and idempotence follow directly from the description of $P_\theta (u,v)$ as the largest common minorant. For associativity, let also $w\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $. A potential lies below $P_\theta (u,v)$ and $w$ if and only if it lies below $u$, $v$ and $w$: when $P_\theta (u,v)\ne -\infty $ this is the maximality of $P_\theta (u,v)$, and otherwise neither condition can be satisfied. Hence $P_\theta (P_\theta (u,v),w)$ and $P_\theta (u,P_\theta (v,w))$ are both the largest common minorant of $u$, $v$ and $w$, or both equal $-\infty $ if there is none. The behavior under constant shifts and decreasing limits follows from proposition 16.11.4, applied to the obstacles $\min \{ u_ j,v_ j\} $, which are bounded above by $\sup _ Xu_1$ and decrease to $\min \{ u,v\} $. If $u$ or $v$ is identically $-\infty $, this limit obstacle has no competitor, in accordance with our convention.

For the last assertion put $B=\max \{ \sup _ Xu,-C\} $. Then $u-B\leq 0$, so $u-B\leq V_\theta $, and also $u-B\leq u+C$. Hence $u-B$ is a common minorant of $u+C$ and $V_\theta $, and