Definition 7.5.1A2CACA24130B

Definition 7.5.1.A2CACA24130B A Borel function $f\colon D\to [-\infty ,\infty ]$ is quasi-continuous (for $\operatorname {Cap}_ D$) if for every compact set $K\subset D$ and every $\varepsilon >0$ there is an open set $G\subset D$ with $\operatorname {Cap}_ D(G)<\varepsilon $ such that the restriction of $f$ to $K\setminus G$ is continuous. Here $[-\infty ,\infty ]$ carries its order topology.