PomeDefinition 7.5.5.DAD5D0809372 A net $(f_ a)$ of Borel functions $f_ a\colon D\to [-\infty ,+\infty ]$ converges locally in capacity to a Borel function $f\colon D\to [-\infty ,+\infty ]$ if, for every compact set $K\subset D$ and every $\delta >0$,
\[ \operatorname {Cap}_ D^*(K\cap \{ f_ a>f+\delta \} )\longrightarrow 0, \qquad \operatorname {Cap}_ D^*(K\cap \{ f_ a<f-\delta \} )\longrightarrow 0. \]
Here $\pm \infty +\delta =\pm \infty $ and $\pm \infty -\delta =\pm \infty $.
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