5.47.19 Countable selections and plurifine locality03B3029EB4B3

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 $. $\square$