Definition 13.7.2019A8D37A03F

Definition 13.7.2.019A8D37A03F For $s,t\in [-\infty ,\infty )$ put $d_*(s,t)=|\arctan s-\arctan t|$, where $\arctan (-\infty )=-\pi /2$. A net $(u_ a)$ of plurisubharmonic functions on a bounded open set $D\subset \mathbb C^ n$ converges in extended capacity to a plurisubharmonic function $u$ if, for every compact set $K\subset D$ and every $\delta >0$, the outer capacity of $K\cap \{ d_*(u_ a,u)>\delta \} $ tends to zero, the capacity being taken as in definition 7.5.5. A net of $\theta $-plurisubharmonic functions on a compact manifold converges in extended capacity if the local plurisubharmonic functions obtained by adding smooth local potentials of $\theta $ do so on the charts of a finite coordinate cover by charts with bounded images. This does not depend on the choice of the local potentials or of the cover, by lemma 13.7.3 below.