5.47 Pluripolar sets and the plurifine topologyF5A3147592B6

Proof. The boundary behavior of $H_ K$ follows from (5.47.3). Since $H_ K$ is bounded, its Monge–Ampère measure is defined (definition 7.2.1).

We first show that $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ vanishes on $B\setminus K$. Choose competitors $v_ j$ for $K$ as in (5.47.4), and let $D$ be a ball with $\overline D\subset B\setminus K$. Let $\widehat v_ j$ be the function obtained from $v_ j$ by corollary 7.7.5: it is plurisubharmonic on $B$, equals $v_ j$ outside $D$, and satisfies $\widehat v_ j\ge v_ j$ and $(\mathrm{dd}^{\mathrm{c}}\widehat v_ j)^ n=0$ on $D$. Each $\widehat v_ j$ is again a competitor for $K$. Indeed, $\widehat v_ j=v_ j\le -1$ on $K$. Moreover $\widehat v_ j\le 0$: outside $D$ because $\widehat v_ j=v_ j$, and on $D$ by the maximum principle (proposition 3.4.7), since by upper semicontinuity the upper limit of $\widehat v_ j$ at each point $x\in \partial D$ is at most $\widehat v_ j(x)=v_ j(x)\le 0$.

Since $\widehat v_ j$ is increasing in $v_ j$ by corollary 7.7.5, the functions $\widehat v_ j$ increase with $j$, and $v_ j\le \widehat v_ j\le h_ K\le H_ K$ shows that $\widehat v_ j\to H_ K$ almost everywhere. They take values in $[-1,0]$, so theorem 7.6.1 gives $(\mathrm{dd}^{\mathrm{c}}\widehat v_ j)^ n\to (\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ weakly, and therefore $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n=0$ on $D$. Covering the open set $B\setminus K$ by countably many such balls $D$, we conclude that $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ is carried by $K$.

As $h_ K=-1$ on $K$, the Borel set $K\cap \{ H_ K>-1\} $ is contained in $\{ h_ K<H_ K\} $, which has outer capacity zero by (5.47.5). Because $H_ K+1$ is a competitor for $\operatorname {Cap}_ B$, this set is therefore null for $(\mathrm{dd}^{\mathrm{c}}(H_ K+1))^ n=(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$. Thus $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ is carried by $K\cap \{ H_ K=-1\} $.

We now prove the capacity formula. As $H_ K+1$ is a competitor for $\operatorname {Cap}_ B$ and $(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ is carried by $K$, we have $\int _ B(\mathrm{dd}^{\mathrm{c}}H_ K)^ n=\int _ K(\mathrm{dd}^{\mathrm{c}}(H_ K+1))^ n\le \operatorname {Cap}_ B(K)$. Conversely, let $v$ be a competitor for $\operatorname {Cap}_ B$, let $0<\varepsilon <1/2$, and put $w=(1-2\varepsilon )(v-1)-\varepsilon $. Then $-1+\varepsilon \le w\le -\varepsilon $. In particular $w>-1$, so $K\cap \{ H_ K\ge w\} \subset K\cap \{ H_ K>-1\} $; this set has outer capacity zero and is therefore null for $(\mathrm{dd}^{\mathrm{c}}w)^ n=(1-2\varepsilon )^ n(\mathrm{dd}^{\mathrm{c}}v)^ n$. Moreover $H_ K(z)\to 0$ as $z\to \partial B$ while $w\le -\varepsilon $, so $\varliminf _{z\to x}(H_ K(z)-w(z))\ge \varepsilon >0$ for every $x\in \partial B$, and the comparison principle (theorem 7.3.11) applies to $H_ K$ and $w$ on $B$. We obtain

Letting $\varepsilon \downarrow 0$ and taking the supremum over $v$, we get $\operatorname {Cap}_ B(K)\le \int _ B(\mathrm{dd}^{\mathrm{c}}H_ K)^ n$.

Now let $(K_ j)$ be compact subsets of $B$ decreasing to $K$. Since $K\subset K_{j+1}\subset K_ j$, the functions $H_{K_ j}$ increase and are bounded above by $H_ K$. Put $H=(\sup _ jH_{K_ j})^*$. By theorem 5.10.2, $H$ is plurisubharmonic and equals $\sup _ jH_{K_ j}$ almost everywhere, and $H\le H_ K$ because $H_ K$ is upper semicontinuous. For the reverse inequality, let $v$ be a competitor for $K$ and let $0<\varepsilon <1$. The open set $G=\{ v<-1+\varepsilon \} $ contains $K$. The compact sets $K_ j\setminus G$ decrease and have empty intersection, so one of them is empty; hence $K_ j\subset G$ for all large $j$. For such $j$ the function $v/(1-\varepsilon )$ is a competitor for $K_ j$, and hence $v\le (1-\varepsilon )H_{K_ j}\le (1-\varepsilon )H$. Letting $\varepsilon \downarrow 0$ gives $v\le H$, and taking the supremum over all competitors $v$ for $K$ gives $h_ K\le H$. As $H$ is upper semicontinuous, $H_ K\le H$. Therefore $H=H_ K$, and $H_{K_ j}\to H_ K$ almost everywhere.

By theorem 7.6.1, $(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n\to (\mathrm{dd}^{\mathrm{c}}H_ K)^ n$ weakly. All these measures are carried by the fixed compact set $K_1$, so testing with a function in $C_ c(B)$ equal to one on $K_1$ shows that their total masses converge. By the capacity formula, $\operatorname {Cap}_ B(K_ j)\to \operatorname {Cap}_ B(K)$, and this convergence is decreasing because capacity is increasing.

Finally, $\operatorname {Cap}_ B(K)\le \operatorname {Cap}_ B^*(K)$ because capacity is increasing. For the converse we may assume $K\neq \varnothing $, since both capacities of the empty set vanish. Choose $\delta >0$ such that the compact sets $K_ j=\{ z:\operatorname {dist}(z,K)\le \delta /j\} $ lie in $B$. They decrease to $K$, and each contains $K$ in its interior. Hence $\operatorname {Cap}_ B^*(K)\le \operatorname {Cap}_ B(\operatorname{Int}K_ j)\le \operatorname {Cap}_ B(K_ j)$, and the right-hand side tends to $\operatorname {Cap}_ B(K)$ by what we have just proved. $\square$

Proof. Choose compact sets $K_ j$ with $K_ j\subset \operatorname{Int}K_{j+1}$ and $\bigcup _ jK_ j=G$. Since the $K_ j$ increase, the functions $H_{K_ j}$ decrease. They take values in $[-1,0]$, so their limit $H$ is plurisubharmonic by theorem 5.10.1. Since $h_{K_ j}=-1$ on $K_ j$ and the upper regularization at a point only involves the values of $h_{K_ j}$ near that point, we have $H_{K_ j}=-1$ on $\operatorname{Int}K_ j$. Every point of $G$ lies in $\operatorname{Int}K_ j$ for all large $j$, so $H=-1$ on $G$. Thus $H$ is a competitor for $G$, and $H\le h_ G$. Conversely, every competitor for $G$ is a competitor for each $K_ j$, hence bounded above by every $H_{K_ j}$ and thus by $H$; so $h_ G\le H$. We have shown $h_ G=H$. As $H$ is upper semicontinuous, $H_ G=h_ G^*=H$; in particular $H_ G=-1$ on $G$. The boundary behavior follows from (5.47.3).

By theorem 7.3.7, $(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n\to (\mathrm{dd}^{\mathrm{c}}H_ G)^ n$ weakly. By proposition 5.47.6, these measures are carried by $K_ j\subset \overline G$. Hence the limit measure vanishes on the open set $B\setminus \overline G$, that is, it is carried by the compact set $\overline G$, and testing with a function in $C_ c(B)$ equal to one on $\overline G$ shows that the total masses converge. Since the Borel sets $K_ j$ increase to $G$, proposition 7.4.3(2) gives $\operatorname {Cap}_ B(K_ j)\to \operatorname {Cap}_ B(G)$. Together with the capacity formula of proposition 5.47.6 for the $K_ j$, this yields $\operatorname {Cap}_ B(G)=\lim _ j\int _ B(\mathrm{dd}^{\mathrm{c}}H_{K_ j})^ n=\int _ B(\mathrm{dd}^{\mathrm{c}}H_ G)^ n$. $\square$

Proof. We prove (2)$\Rightarrow $(1), then (1)$\Rightarrow $(3) for relatively compact sets $E$ and (3)$\Rightarrow $(2), deduce (1)$\Rightarrow $(2) in general by exhaustion, and finally prove (2)$\Rightarrow $(3). The remaining assertions then follow by localization.

(2)$\Rightarrow $(1). We show more generally that every pluripolar set $E\subset B$ satisfies (1); a set as in (2) is pluripolar. By definition 5.47.2, every point of $B$ has an open neighborhood $W\subset B$ carrying a plurisubharmonic function whose pole set contains $E\cap W$. The second assertion of proposition 7.4.4, applied with $D=B$, therefore gives $\operatorname {Cap}_ B^*(E\cap L)=0$ for every compact set $L\subset B$. Exhausting $B$ by countably many compact sets and using countable subadditivity, we obtain $\operatorname {Cap}_ B^*(E)=0$.

(1)$\Rightarrow $(3) when $E\Subset B$. Fix an open set $O$ with $\overline E\subset O\Subset B$. Since $\operatorname {Cap}_ B^*(E)=0$, there are open sets $G_ j'\supset E$ with $\operatorname {Cap}_ B(G_ j')<1/j$. The open sets $G_ j=O\cap G_1'\cap \cdots \cap G_ j'$ decrease, contain $E$, lie in $O$, and satisfy $\operatorname {Cap}_ B(G_ j)\to 0$. Since the $G_ j$ decrease, the functions $H_{G_ j}$ increase; put $H=(\sup _ jH_{G_ j})^*$. By theorem 5.10.2, $H$ is plurisubharmonic, takes values in $[-1,0]$, and is the almost-everywhere limit of the $H_{G_ j}$. By proposition 5.47.7, $\int _ B(\mathrm{dd}^{\mathrm{c}}H_{G_ j})^ n=\operatorname {Cap}_ B(G_ j)\to 0$, and by theorem 7.6.1 these measures converge weakly to $(\mathrm{dd}^{\mathrm{c}}H)^ n$. Hence $\int \chi \, (\mathrm{dd}^{\mathrm{c}}H)^ n=\lim _ j\int \chi \, (\mathrm{dd}^{\mathrm{c}}H_{G_ j})^ n\le \lim _ j\operatorname {Cap}_ B(G_ j)=0$ for every $\chi \in C_ c(B)$ with $0\le \chi \le 1$, and therefore $(\mathrm{dd}^{\mathrm{c}}H)^ n=0$.

Since every $G_ j$ lies in $O$, the barrier (5.47.3), with a constant $A$ chosen for $O$, bounds every $H_{G_ j}$, and hence $H$, from below by one function tending to zero at $\partial B$; thus $H(z)\to 0$ as $z\to \partial B$. The last assertion of the comparison principle (theorem 7.3.11), applied to $u=H$ and $v=0$, gives $H\ge 0$, and therefore $H=0$. Finally, $E\subset G_ j$ gives $H_{G_ j}\le H_ E$ for all $j$; since $H_ E$ is upper semicontinuous, $0=H\le H_ E\le 0$.

(3)$\Rightarrow $(2). Suppose $H_ E=0$, and choose competitors $v_ j$ for $E$ as in (5.47.4). Fix a point $a\in B$ with $v_ j(a)\to 0$, and choose indices $j_1<j_2<\cdots $ with $v_{j_ l}(a)>-2^{-l}$. Since every term is nonpositive, the partial sums of $V=\sum _ lv_{j_ l}$ form a decreasing sequence of nonpositive plurisubharmonic functions, and $V(a)\ge -1$. By theorem 5.10.1, $V$ is plurisubharmonic on $B$; moreover $V\le 0$, and $V=-\infty $ on $E$ because every term is at most $-1$ there.

(1)$\Rightarrow $(2) in general. Let $\operatorname {Cap}_ B^*(E)=0$, and let $(L_ l)$ be an exhaustion of $B$ by compact sets. Each set $E\cap L_ l$ has outer capacity zero and is relatively compact in $B$, so the two preceding steps give $V_ l\in \mathrm{PSH}(B)$ with $V_ l\le 0$ and $E\cap L_ l\subset \{ V_ l=-\infty \} $. The pole sets of the $V_ l$ are Lebesgue null, so there is a point $a\in B$ at which all $V_ l$ are finite. Choose constants $c_ l>0$ with $c_ l|V_ l(a)|\le 2^{-l}$. The partial sums of $\sum _ lc_ lV_ l$ decrease and are at least $-1$ at $a$, so, as in the previous step, $V=\sum _ lc_ lV_ l$ is plurisubharmonic by theorem 5.10.1, and it is nonpositive. Every point of $E$ lies in some $L_ l$, where $c_ lV_ l=-\infty $ while all other terms are nonpositive; hence $V=-\infty $ on $E$.

(2)$\Rightarrow $(3). If $V$ is as in (2), then $\varepsilon V$ is a competitor for $E$ for every $\varepsilon >0$, so $\varepsilon V\le h_ E\le 0$. Letting $\varepsilon \downarrow 0$, we get $h_ E=0$ on the set $\{ V>-\infty \} $. This set has full measure and is therefore dense in $B$, so every neighborhood of a point contains points where $h_ E=0$. Hence $H_ E=h_ E^*\ge 0$, and $H_ E=0$. This completes the proof of the equivalences.

By the proof of (2)$\Rightarrow $(1), every pluripolar subset of $B$ satisfies (1), and hence all three conditions; in particular, it is globally pluripolar in $B$. Now let $X$ be a complex manifold and $E=\bigcup _ mE_ m$, where each $E_ m\subset X$ is pluripolar. Let $x\in X$, and let $B$ be a coordinate ball around $x$, identified with an open ball in $\mathbb C^ n$. Each $E_ m\cap B$ is pluripolar in $B$, so $\operatorname {Cap}_ B^*(E_ m\cap B)=0$ for every $m$, and $\operatorname {Cap}_ B^*(E\cap B)=0$ by countable subadditivity. By (1)$\Rightarrow $(2), the set $E\cap B$ lies in the pole set of a plurisubharmonic function on the connected coordinate neighborhood $B$ of $x$. Hence $E$ is pluripolar.

Finally, let $W\subset X$ be open and let $E\subset W$ be pluripolar in $W$. That $E$ is pluripolar in $X$ is not immediate from definition 5.47.2 at the boundary points of $W$, where a single plurisubharmonic function on a neighborhood must have poles at all nearby points of $E$. Let $x\in X$, and let $U$ be a coordinate ball around $x$, identified with an open ball in $\mathbb C^ n$. The open set $W\cap U$ is a countable union of balls $D_ k$ with $\overline{D_ k}\subset W\cap U$. Each set $E\cap D_ k$ is pluripolar in $U$: at points of $\overline{D_ k}\subset W$ we may use, after restriction to small balls in $U$, the functions provided by the pluripolarity of $E$ in $W$, and every point of $U\setminus \overline{D_ k}$ has a neighborhood disjoint from $D_ k$. By the previous paragraph, applied in $U$, the countable union $E\cap U=\bigcup _ k(E\cap D_ k)$ is pluripolar in $U$, and hence, by the first part of the theorem, it lies in the pole set of a plurisubharmonic function on $U$. Thus $E$ is pluripolar in $X$. $\square$

Proof. Step 1: the glued functions. Choose compact sets $L_1,L_2,\ldots $ whose interiors cover $\mathbb C^ n$, put $E_ m=E\cap L_ m$, and choose a sequence $(m_ j)_{j\ge 1}$ in which every positive integer occurs infinitely often. Since each $L_ m$ is bounded, we can choose radii $R_ j>1$ increasing to $\infty $ such that

in particular $L_{m_ j}$ lies in the ball $B_ j=B(0,R_ j)$. The set $E\cap B_ j$ is pluripolar in $B_ j$, so by theorem 5.47.8 the relative extremal function of its subset $E_{m_ j}$ with respect to $B_ j$ vanishes identically. The competitors of (5.47.4) for $E_{m_ j}$ in $B_ j$ therefore converge to zero almost everywhere on $B_ j$, and by bounded convergence their integrals over $B_ j$ tend to zero. Since a competitor for $E_{m_ j}$ with values in $[-1,0]$ equals $-1$ on $E_{m_ j}$, one of these competitors, which we call $u_ j$, satisfies

The function $\log ^+|z|=\max \{ \tfrac 12\log |z|^2,0\} $ is plurisubharmonic on $\mathbb C^ n$ by proposition 5.10.3, proposition 5.9.5, and so is $\rho _ j$. On $\partial B_ j$ we have $|z|=R_ j>1$, hence $\rho _ j=0$, while the upper limit of $u_ j$ at every point of $\partial B_ j$ is at most $0$. By proposition 5.9.5, the function $V_ j$ equal to $\max \{ u_ j,\rho _ j\} $ on $B_ j$ and to $\rho _ j$ on $\mathbb C^ n\setminus B_ j$ is plurisubharmonic on $\mathbb C^ n$. Since $\rho _ j\le -1=u_ j$ on $E_{m_ j}$, we have $V_ j=-1$ on $E_{m_ j}$. On $B_ j$ both $u_ j$ and $\rho _ j$ are nonpositive, so $V_ j\le 0$ there, while $V_ j=\rho _ j$ outside $B_ j$; hence

In particular $V_ j$ is locally bounded.

Step 2: the sum. Fix $r>0$ and choose $j_0$ with $R_{j_0}>r$. For $j\ge j_0$ the ball $B(0,r)$ lies in $B_ j$, so $V_ j\le 0$ on $B(0,r)$, and since $V_ j\ge u_ j$ there,

Hence the partial sums of $\sum _{j\ge j_0}V_ j$ decrease on $B(0,r)$, and by monotone convergence their limit $W_ r$ satisfies $\int _{B(0,r)}(-W_ r)\, \mathrm dV_{2n}\le \sum _ j2^{-j}<\infty $. Thus $W_ r$ is not identically $-\infty $, and $W_ r\in \mathrm{PSH}(B(0,r))$ by theorem 5.10.1. At every point of $\mathbb C^ n$ all but finitely many terms of the series $V=\sum _ jV_ j$ are nonpositive, so $V$ is well defined with values in $[-\infty ,\infty )$. On $B(0,r)$ it equals $\sum _{j<j_0}V_ j+W_ r$, a sum of finitely many finite plurisubharmonic functions and a plurisubharmonic function; hence $V$ is plurisubharmonic on $B(0,r)$. As $r$ is arbitrary and plurisubharmonicity is a local property, $V\in \mathrm{PSH}(\mathbb C^ n)$.

Let $z\in E$, so that $z\in E_ m$ for some $m$. Since $m=m_ j$ for infinitely many $j$, infinitely many terms $V_ j(z)$ equal $-1$, while all terms with $R_ j>|z|$ are nonpositive. Hence $V(z)=-\infty $. Finally, summing the upper bounds of Step 1 gives $V\le \sum _ j2^{-j}\log ^+|z|=\log ^+|z|$. $\square$

Proof. (1) Let $x\in X$, and let $B$ be a coordinate ball around $x$, identified with an open ball in $\mathbb C^ n$. On $B$ the regularization $F^*$ is the regularization of $F|_ B$, so by corollary 7.6.2 the set $L\cap \{ F<F^*\} $ has outer capacity zero in $B$ for every compact set $L\subset B$. Exhausting $B$ by countably many compact sets, we obtain $\operatorname {Cap}_ B^*(B\cap \{ F<F^*\} )=0$. By theorem 5.47.8, $B\cap \{ F<F^*\} $ lies in the pole set of a plurisubharmonic function on $B$. Hence $\{ F<F^*\} $ is pluripolar.

(2) For an increasing sequence, $\lim _ ju_ j=\sup _ ju_ j$, so this is the special case of (1) for the family $(u_ j)$.

(3) Let $E$ be pluripolar, and work in a chart, identified with an open set $\Omega \subset \mathbb C^ n$. Let $B'\Subset B\Subset \Omega $ be concentric open balls. The set $E\cap B$ is pluripolar in $B$, so $\operatorname {Cap}_ B^*(E\cap B)=0$ by theorem 5.47.8.

Let $\mu $ be the trace measure of the product, or the total variation of one of its coefficient measures. The restrictions of $u_1,\ldots ,u_ q$ to $B$ form a family that is uniformly bounded on each compact subset of $B$, since $\overline B\subset \Omega $ is compact; so lemma 7.5.7(1), applied to this family and the compact set $\overline{B'}$, gives a constant $C$ with $\mu (A)\le C\operatorname {Cap}_ B(A)$ for every Borel set $A\subset \overline{B'}$. Given $\varepsilon >0$, choose an open set $G\supset E\cap B$ with $\operatorname {Cap}_ B(G)<\varepsilon $; then the outer measure of $E\cap B'$ satisfies $\mu ^*(E\cap B')\le \mu (G\cap B')\le C\operatorname {Cap}_ B(G\cap B')\le C\operatorname {Cap}_ B(G)<C\varepsilon $. Hence $\mu ^*(E\cap B')=0$. Since $\Omega $ is covered by countably many such balls $B'$, the set $E\cap \Omega $ has outer measure zero for $\mu $. For a Borel set $E$ this means that the product does not charge $E$. $\square$

Proof. We begin with the first two assertions of (3). Let $W\subset X$ be open. Every subbasic set of the plurifine topology of $W$ is a subbasic set of the plurifine topology of $X$. Conversely, the intersection of a subbasic set of $X$ with $W$ is subbasic for $W$, because the restriction of a function $u\in \mathrm{PSH}(V)$ to $V\cap W$ is plurisubharmonic. Since the sets $S\cap W$, with $S$ subbasic for $X$, form a subbasis of the induced topology, the plurifine topology of $W$ is the induced topology.

Next, if $F$ is a biholomorphism, then $F^{-1}(\{ u>t\} )=\{ u\circ F>t\} $, and $u\circ F$ is plurisubharmonic whenever $u$ is. Indeed, by proposition 5.9.3, in connected charts $u\circ F$ is plurisubharmonic or identically $-\infty $, and the second alternative would make $u$ identically $-\infty $ on a nonempty open set, which is impossible because $u$ is locally integrable (theorem 5.9.1). Applying this to $F$ and to $F^{-1}$, we see that both map subbasic sets to subbasic sets. In particular, we may compute plurifine neighborhoods of $a$ in a chart, as if $X$ were an open subset of $\mathbb C^ n$.

(2) Each set $B\cap \{ \psi >0\} $ as in (2) is plurifine open, as the intersection of a Euclidean open set with a subbasic set. Now let $N$ be a plurifine neighborhood of $a$. By the definition of the topology, $N$ contains a set

containing $a$, where $W_0$ is a Euclidean open neighborhood of $a$, $k\ge 0$, each $u_ i$ is plurisubharmonic on a Euclidean open neighborhood $V_ i$ of $a$, and $c_ i\in \mathbb R$. If $k=0$, every coordinate ball $B\subset W_0$ centered at $a$ works, with $\psi =1$. Let $k\ge 1$. Replacing $u_ i$ by $\max \{ u_ i,c_ i-1\} $, which does not change $\{ u_ i>c_ i\} $, we may assume that the $u_ i$ are locally bounded. Put $\delta _ i=u_ i(a)-c_ i>0$. By upper semicontinuity there is a coordinate ball $B'$ centered at $a$, contained in $W_0$ and in every $V_ i$, on which every $u_ i$ is bounded and satisfies $u_ i<u_ i(a)+\delta _ i/(2k)$. Let $B$ be any coordinate ball centered at $a$ with $\overline B\subset B'$, and put

This is a bounded plurisubharmonic function on $B'$ with $\psi (a)=1/4>0$. If $z\in B$ and $u_ i(z)\le c_ i$ for some $i$, then the $i$-th term of the sum is at most $-1$ and each of the other $k-1$ terms is less than $1/(2k)$, so $\psi (z)\le 1/4-1+(k-1)/(2k)<0$. Hence $B\cap \{ \psi >0\} \subset N_0\subset N$, which proves (2).

(1) Let $U$ be a plurifine neighborhood of $a$, and put $E=X\setminus U$; note that $a\notin E$. By (2), $U$ contains a set $B\cap \{ \psi >0\} $ with $\psi (a)>0$, so $\psi \le 0$ on $E\cap B$. Hence either $a$ is not in the Euclidean closure of $E$, or $\varlimsup _{E\ni z\to a}\psi (z)\le 0<\psi (a)$; in both cases $E$ is plurithin at $a$. Conversely, suppose that $E=X\setminus U$ is plurithin at $a\in U$. If $a$ is not in the Euclidean closure of $E$, then $U$ contains a Euclidean neighborhood of $a$. Otherwise let $\psi $ be as in definition 5.47.14, and choose $c$ with $\varlimsup _{E\ni z\to a}\psi (z)<c<\psi (a)$. Then $\psi <c$ on $E\cap B$ for a small ball $B$ centered at $a$, so the plurifine open set $B\cap \{ \psi >c\} $ contains $a$ and is contained in $U$.

(3) It remains to prove that a holomorphic map $F\colon Y\to X$ is continuous for the plurifine topologies. Preimages of Euclidean open sets are Euclidean open. Let $u\in \mathrm{PSH}(V)$ with $V\subset X$ open, let $t\in \mathbb R$, and let $y\in F^{-1}(\{ u>t\} )$. Choose a connected coordinate neighborhood $Y'$ of $y$ such that $F(Y')$ lies in a connected coordinate neighborhood contained in $V$. By proposition 5.9.3, $u\circ F$ is either identically $-\infty $ or plurisubharmonic on $Y'$; the first alternative is excluded because $u(F(y))>t$. Thus $Y'\cap \{ u\circ F>t\} $ is a plurifine open neighborhood of $y$ contained in $F^{-1}(\{ u>t\} )$. Hence preimages of subbasic sets are plurifine open. $\square$

Proof. (1) Let $O$ be a nonempty plurifine open set and $a\in O$. By proposition 5.47.15(2), $O$ contains a set $A=B\cap \{ \psi >0\} $ with $a\in A$, where $B$ is a coordinate ball and $\psi $ is a bounded plurisubharmonic function on a neighborhood of $\overline B$. This set is Borel because $\psi $ is upper semicontinuous. If $A$ had measure zero, then $\max \{ \psi ,0\} =0$ almost everywhere on $B$; since both functions are plurisubharmonic on $B$, they would coincide everywhere by the uniqueness in theorem 5.9.1, contradicting $\psi (a)>0$.

(2) A pluripolar set is Lebesgue null in every chart, so by (1) it contains no nonempty plurifine open set. Equivalently, its complement meets every nonempty plurifine open set.

(3) The set $\{ u>-\infty \} =\bigcup _ m\{ u>-m\} $ is a union of subbasic plurifine open sets. $\square$

Proof. Since $X$ has a countable basis, it is covered by countably many charts; in each chart, the coordinate balls with rational centers and radii and with closure in the chart domain form a countable basis. Let $\mathcal Q$ be the union of these countable families; it is a countable basis of $X$.

(1) Let $U=\bigcup _ iU_ i$; if $I$ is empty, take $J$ empty. We call a set basic if it has the form $Q\cap \{ \psi >0\} $, where $Q\in \mathcal Q$ and $\psi $ is a bounded plurisubharmonic function on a neighborhood of $\overline Q$. Every point $a\in U$ lies in a basic set contained in some $U_ i$: if $a\in U_ i$, then proposition 5.47.15(2) gives a ball $B$ and a bounded plurisubharmonic function $\psi $ on a neighborhood of $\overline B$ with $a\in B\cap \{ \psi >0\} \subset U_ i$, and it suffices to choose $Q\in \mathcal Q$ with $a\in Q\subset B$.

Fix $Q\in \mathcal Q$, and let $\Psi _ Q$ be the set of bounded plurisubharmonic functions $\psi $ on neighborhoods of $\overline Q$ for which the basic set $Q\cap \{ \psi >0\} $ is contained in some $U_ i$. If $\Psi _ Q$ is empty, we put $P_ Q=\varnothing $ and select nothing for this $Q$. Otherwise, we may assume that all members of $\Psi _ Q$ take values in $[-1,1]$ on $Q$, since replacing $\psi $ by $\psi /c$, where $c>0$ bounds $|\psi |$ on $Q$, does not change $Q\cap \{ \psi >0\} $.

Let $F_ Q$ be the pointwise supremum of the members of $\Psi _ Q$ on $Q$. By Choquet's lemma (lemma 3.4.22) there are $\psi _{Q,1},\psi _{Q,2},\ldots $ in $\Psi _ Q$ whose supremum $F_ Q'$ satisfies $(F_ Q')^*=F_ Q^*$. By corollary 5.47.11(1) and the last assertion of theorem 5.47.8, the set $P_ Q=\{ F_ Q<F_ Q^*\} \cup \{ F_ Q'<(F_ Q')^*\} $ is pluripolar in $Q$, and hence in $X$. If $z\in Q\setminus P_ Q$ and $\psi (z)>0$ for some $\psi \in \Psi _ Q$, then $F_ Q'(z)=F_ Q^*(z)=F_ Q(z)>0$, so $\psi _{Q,k}(z)>0$ for some $k$. Therefore

For each $Q$ and $k$ choose an index $i(Q,k)\in I$ with $Q\cap \{ \psi _{Q,k}>0\} \subset U_{i(Q,k)}$, and let $J$ be the countable set of these indices. Every point of $U$ lies in some basic set $Q\cap \{ \psi >0\} $ with $\psi \in \Psi _ Q$, so $U\setminus \bigcup _{j\in J}U_ j$ is contained in $\bigcup _ QP_ Q$. This countable union of pluripolar sets is pluripolar by theorem 5.47.8.

(2) Apply the construction of (1) to the family consisting of the single set $U$. It selects countably many sets $Q\cap \{ \psi _{Q,k}>0\} $, where $Q\in \mathcal Q$ is a coordinate ball and $\psi _{Q,k}$ restricts to a bounded plurisubharmonic function on $Q$; these sets are contained in $U$, and their union $V$ satisfies $U\setminus V\subset \bigcup _ QP_ Q$, which is pluripolar. Each selected set is Borel, because $\psi _{Q,k}$ is upper semicontinuous, so the countable union $V$ is Borel. Finally, let $C$ and $\mu $ be as in the statement. The set $(U\setminus V)\cap C$ is pluripolar in $C$, so it has outer $\mu $-measure zero and is a null set for the completion of $\mu $. Since $U\cap C=(V\cap C)\cup \bigl ((U\setminus V)\cap C\bigr )$ and $V\cap C$ is Borel, $U\cap C$ is measurable for the completion of $\mu $. $\square$

Proof. Replacing $t$ by $\max \{ t,0\} $, which is plurisubharmonic (proposition 5.9.5) and has the same set $\{ t>0\} $, we may assume that $t\ge 0$. Then $t$ is locally bounded, because plurisubharmonic functions are bounded above on compact sets, and $u+\varepsilon t$ is a locally bounded plurisubharmonic function for every $\varepsilon >0$. For $\varepsilon >0$ put $w_\varepsilon =\max \{ u+\varepsilon t,v\} $. On $\{ t>0\} $ we have $u+\varepsilon t>u=v$, so $\{ t>0\} \subset \{ u+\varepsilon t>v\} $, and proposition 7.3.10, applied to $u+\varepsilon t$ and $v$, gives

Multiplying by $t$, which vanishes outside $\{ t>0\} $, we obtain $t(\mathrm{dd}^{\mathrm{c}}w_\varepsilon )^ p\wedge T=t(\mathrm{dd}^{\mathrm{c}}(u+\varepsilon t))^ p\wedge T$. As $\varepsilon $ decreases to zero along a sequence, $w_\varepsilon $ decreases to $\max \{ u,v\} $ and $u+\varepsilon t$ decreases to $u$, because $t\ge 0$. The weighted form of theorem 7.3.7, with the constant sequence of weights equal to $t$, therefore yields

Interchanging the roles of $u$ and $v$, we also get $t(\mathrm{dd}^{\mathrm{c}}\max \{ u,v\} )^ p\wedge T=t(\mathrm{dd}^{\mathrm{c}}v)^ p\wedge T$, and hence $t(\mathrm{dd}^{\mathrm{c}}u)^ p\wedge T=t(\mathrm{dd}^{\mathrm{c}}v)^ p\wedge T$. Finally, multiply the coefficient measures of both sides by $1/(t+\delta )$ and let $\delta \downarrow 0$. Since $t/(t+\delta )\to \mathbf1_{\{ t>0\} }$ pointwise and boundedly, dominated convergence with respect to the total variations of the coefficient measures gives the assertion. $\square$

Proof. The identity and the measurability of $W$ are local properties, so we may assume that $X$ is an open ball in $\mathbb C^ n$; by proposition 5.47.15(3), the intersection of $W$ with this ball is plurifine open in the ball. Write $S_ u=\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$ and $S_ v=\mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q$. We first prove the identity on sets of the form $Q\cap \{ \psi >0\} $, and then cover $W$ by countably many such sets up to a pluripolar set, which the products do not charge.

Step 1: basic sets. We show that $\mathbf1_ AS_ u=\mathbf1_ AS_ v$ whenever $Q\subset X$ is open, $\psi \in \mathrm{PSH}(Q)$ and $A=Q\cap \{ \psi >0\} \subset W$. For every nonempty set $I\subset \{ 1,\ldots ,q\} $, the functions $u_ I=\sum _{l\in I}u_ l$ and $v_ I=\sum _{l\in I}v_ l$ are locally bounded and plurisubharmonic on $Q$, and they coincide on $A$, because $u_ l=v_ l$ on $W$. By lemma 5.47.21, applied on $Q$ with $t=\psi $, $T=1$ and $p=q$, we have $\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}u_ I)^ q=\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}v_ I)^ q$.

We now recover $S_ u$ from the products $(\mathrm{dd}^{\mathrm{c}}u_ I)^ q$ by polarization. For a multi-index $\alpha \in \mathbb N^ q$ with $|\alpha |=q$, let $S_ u^\alpha $ be the product of $q$ factors in which $\mathrm{dd}^{\mathrm{c}}u_ l$ occurs $\alpha _ l$ times; it does not depend on the order of the factors, because products of locally bounded plurisubharmonic functions are symmetric (theorem 7.3.7). Since these products are also additive in each factor (proposition 7.2.2(3)), expanding $(\mathrm{dd}^{\mathrm{c}}u_ I)^ q$ with $\mathrm{dd}^{\mathrm{c}}u_ I=\sum _{l\in I}\mathrm{dd}^{\mathrm{c}}u_ l$ gives, exactly as in the multinomial expansion of $(\sum _{l\in I}x_ l)^ q$,

where $\alpha $ ranges over the multi-indices with $|\alpha |=q$ and $\alpha _ l=0$ for $l\notin I$. Multiplying by $(-1)^{q-|I|}$ and summing over the nonempty sets $I\subset \{ 1,\ldots ,q\} $, we obtain $\sum _\alpha c_\alpha S_ u^\alpha $, summed over all $\alpha $ with $|\alpha |=q$, where $c_\alpha $ is the coefficient of the monomial $x^\alpha $ on the right-hand side of the polarization identity (7.3.9) with $N=q$ (in which the term with $I=\varnothing $ vanishes). Both sides of that identity are polynomials in $x_1,\ldots ,x_ q$ that agree on $\mathbb R^ q$, so they have the same coefficients: $c_\alpha =q!$ if $\alpha =(1,\ldots ,1)$, and $c_\alpha =0$ otherwise. Since $S_ u^{(1,\ldots ,1)}=S_ u$, we obtain

The same identity holds for the $v_ l$. Multiplying both identities by $\mathbf1_ A$ and using $\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}u_ I)^ q=\mathbf1_ A(\mathrm{dd}^{\mathrm{c}}v_ I)^ q$, we obtain $\mathbf1_ AS_ u=\mathbf1_ AS_ v$.

Step 2: countable selection. By theorem 5.47.20(2), applied to the ball $X$ and its plurifine open subset $W$, there are countably many sets $A_ k=Q_ k\cap \{ \psi _ k>0\} \subset W$, with $Q_ k\subset X$ open and $\psi _ k\in \mathrm{PSH}(Q_ k)$, whose union $V$ is Borel and such that $W\setminus V$ is pluripolar. By corollary 5.47.11(3), every pluripolar subset of $X$ has outer measure zero for the total variations of the coefficient measures of $S_ u$ and $S_ v$. Hence, by the last assertion of theorem 5.47.20(2), $W$ is measurable for their completions; moreover $W\setminus V$ is a null set for them, so $\mathbf1_ WS_ u=\mathbf1_ VS_ u$ and $\mathbf1_ WS_ v=\mathbf1_ VS_ v$. The Borel sets $A_ k'=A_ k\setminus (A_1\cup \cdots \cup A_{k-1})$ partition $V$, and Step 1 gives $\mathbf1_{A_ k'}S_ u=\mathbf1_{A_ k'}\mathbf1_{A_ k}S_ u=\mathbf1_{A_ k'}\mathbf1_{A_ k}S_ v=\mathbf1_{A_ k'}S_ v$. Summing over $k$, by countable additivity of the coefficient measures, we obtain $\mathbf1_ VS_ u=\mathbf1_ VS_ v$, which proves the theorem. $\square$

Proof. Each set $\{ u_ l>-m\} $ is a subbasic plurifine open set, so $E_ m$ is plurifine open; it is also Borel. On $E_ m$ we have $u_ l>-m$, and hence $u_ l^{(k)}=u_ l=u_ l^{(m)}$ for all $l$ and all $k\ge m$. The truncations are locally bounded plurisubharmonic functions, so theorem 5.47.22 gives $\mathbf1_{E_ m}S_ k=\mathbf1_{E_ m}S_ m$. Taking trace measures, and using $E_ m\subset E_ k$ and the positivity of $\mu _{S_ k}$, we obtain $\mu _ m=\mathbf1_{E_ m}\mu _{S_ k}\le \mathbf1_{E_ k}\mu _{S_ k}=\mu _ k$. By corollary 5.47.11(3), every pluripolar subset of $\Omega $ has outer measure zero for $\mu _{S_ m}$, and hence for $\mu _ m$.

For a Borel set $A\subset \Omega $ put $\mu (A)=\lim _ m\mu _ m(A)\in [0,\infty ]$. If $A$ is the disjoint union of Borel sets $A_ i$, then

by monotone convergence for series of nonnegative terms that increase with $m$. Thus $\mu $ is a Borel measure. Finally, let $N\subset \Omega $ be pluripolar. Since $N$ has outer measure zero for each $\mu _ m$, there are Borel sets $G_{m,i}\supset N$ with $\mu _ m(G_{m,i})<1/i$ for all $m,i\ge 1$. The Borel set $G=\bigcap _{m,i}G_{m,i}$ contains $N$ and satisfies $\mu _ m(G)=0$ for every $m$, hence $\mu (G)=0$. Thus $N$ has outer measure zero for $\mu $. $\square$

Proof. The function $\varphi $ is plurifinely continuous with values in $[-\infty ,\infty )$, and $\{ -\infty \} $ is closed in $[-\infty ,\infty )$; hence the pole set $\{ \varphi =-\infty \} $ is plurifine closed. The same holds for $\psi $, so $P$ is plurifine closed. It is pluripolar because it lies in the pole set of $\varphi $, which is pluripolar by definition 5.47.2, applied with small balls in $\Omega $ as neighborhoods.

The subtraction $(s,t)\mapsto s-t$ is continuous from $[-\infty ,\infty )^2\setminus \{ (-\infty ,-\infty )\} $ to $[-\infty ,\infty ]$. At points where $s$ and $t$ are finite, this is the continuity of subtraction on $\mathbb R^2$. If $s=-\infty $ and $t$ is finite, then for every $M>0$ we have $s'-t'<-M$ whenever $t'>t-1$ and $s'<t-1-M$; this is continuity with the value $-\infty $. Symmetrically, if $s$ is finite and $t=-\infty $, then $s'-t'>M$ whenever $s'>s-1$ and $t'<s-1-M$. Since $z\mapsto (\varphi (z),\psi (z))$ is plurifinely continuous on $\Omega \setminus P$ with values in this domain, $\varphi -\psi $ is plurifinely continuous on $\Omega \setminus P$. Finally, $b$ is continuous for the Euclidean topology, hence for the plurifine topology, and addition is continuous from $[-\infty ,\infty ]\times \mathbb R$ to $[-\infty ,\infty ]$. $\square$

Proof. The set $W'=W\setminus P$ is plurifine open, and $f$ is plurifinely continuous on $W'$, by proposition 5.47.25. Let $N$ be a pluripolar set containing $P$. We claim that $\sup _{W'\setminus N}f=\sup _{W'}f$. The inequality $\le $ holds because $W'\setminus N\subset W'$. Conversely, if $s\in \mathbb R$ and $s<\sup _{W'}f$, then $\{ z\in W':f(z)>s\} $ is a nonempty plurifine open set. It meets $\Omega \setminus N$, which is plurifine dense by proposition 5.47.16(2), and hence $\sup _{W'\setminus N}f>s$. This proves the claim, and the same argument with the sets $\{ f<s\} $ gives $\inf _{W'\setminus N}f=\inf _{W'}f$.

Since $W\setminus N=W'\setminus N$ whenever $N\supset P$, every pluripolar set $N\supset Z$ gives $\sup _{W\setminus N}f=\sup _{W\setminus P}f$; taking $N=Z$, we also get $\sup _{W\setminus Z}f=\sup _{W\setminus P}f$. Hence $\operatorname *{q-sup}_ Wf=\sup _{W\setminus Z}f=\sup _{W\setminus P}f$, and similarly $\operatorname *{q-inf}_ Wf=\inf _{W\setminus Z}f=\inf _{W\setminus P}f$. The common values do not involve $Z$. $\square$