Lemma 16.11.133A40EB62C98

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 $\operatorname {Cap}_ D^*(E\cap D)=0$. $\square$