PomeDefinition 7.4.2.E9E7C75396D2 Let $D\subset \mathbb {C}^ n$, $n\ge 1$, be a bounded open set. The Monge–Ampère capacity of a Borel set $E\subset D$ is
\[ \operatorname {Cap}_ D(E)=\sup \left\{ \int _ E(\mathrm{dd}^{\mathrm{c}}v)^ n: v\in \mathrm{PSH}(D),\ 0\le v\le 1\right\} . \]
The functions $v$ in this supremum are called competitors for $\operatorname {Cap}_ D$. The outer capacity of an arbitrary set $A\subset D$ is
\[ \operatorname {Cap}_ D^*(A)=\inf \{ \operatorname {Cap}_ D(G):A\subset G\subset D, \ G\text{ open}\} . \]
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