Lemma 5.29.18D0C7DB81F772

Proof. If the first assertion failed, there would be $\epsilon >0$ and Borel sets $E_ j$ of capacity at most $2^{-j}$ with $\mu (E_ j)\geq \epsilon $. By countable subadditivity of capacity,

Hence $\limsup _ jE_ j$ is pluripolar and has $\mu $-measure zero. On the other hand, since $\mu $ is finite, continuity from above applied to the sets $\bigcup _{k\geq j}E_ k$ shows that $\limsup _ jE_ j$ has measure at least $\epsilon $, a contradiction. If $u_ j\to u$ in capacity, applying the first assertion to the sets $\{ |u_ j-u|>\delta \} $ gives the consequence. $\square$