Lemma 5.29.15DF96BD63B553

Proof. Choose open sets $O_ j$ of capacity at most $2^{-j}$ such that the restriction of $f$ to $X\setminus O_ j$ is continuous, and set $G_ j=\bigcup _{k\geq j}O_ k$. Then the sets $G_ j$ decrease, their capacities tend to zero, and $f$ is continuous on the closed complement of each $G_ j$. Choose $M\geq \sup _ Xf$ and define

At a boundary point of $G_ j$, the values of $f_ j$ on the closed complement tend to the value of $f$ there, while the values inside $G_ j$ are at least that value. Thus $f_ j$ is lower semicontinuous. The functions $f_ j$ decrease to $f$ outside $\bigcap _ jG_ j$, a set of capacity zero. Since $f_ j-f$ is supported in $G_ j$, they also converge to $f$ in capacity. $\square$