Lemma 7.11.27A2369B2DB317

Proof. Let $C$ be a common bound for $|f_ j|$ and $|f|$. Since $\omega $ is nonnegative and nondecreasing with $\omega (t)\to 0$ as $t\downarrow 0$, we have $\omega (0)=0$ and $\omega (t_ j)\to 0$ whenever $t_ j\geq 0$ and $t_ j\to 0$. Choose $\chi \in C_ c(B)$ with $0\leq \chi \leq 1$ and $\chi =1$ on $K_0$. Since the measures are carried by $K_0$, we have $\nu _ j(B)=\int \chi \, \mathrm d\nu _ j\to \int \chi \, \mathrm d\nu =\nu (B)$, so the masses of the $\nu _ j$ and of $\nu $ have a common bound $M$. The capacity bound passes to $\nu $ on open sets: if $G\subset B$ is open, then $\operatorname {Cap}^*_ B(G)=\operatorname {Cap}_ B(G)$, and proposition 2.6.2 gives $\nu (G)\leq \varliminf _ j\nu _ j(G)\leq \omega (\operatorname {Cap}_ B(G))$.

We first consider the fixed weight $f$. Let $L\subset B$ be a compact neighborhood of $K_0$ on which $f$ is quasi-continuous, and let $\kappa >0$. Choose an open set $G\subset B$ with $\operatorname {Cap}_ B(G)<\kappa $ such that $f$ is continuous on the compact set $L\setminus G$. By lemma 7.5.3, this restriction extends to a continuous function on $\mathbb C^ n$ bounded by $C$. Multiplying the extension by a cutoff in $C_ c(\operatorname{Int}L)$ with values in $[0,1]$ and equal to one on $K_0$, we obtain $g\in C_ c(B)$ with $|g|\leq C$ and $g=f$ on $K_0\setminus G$. Since $\nu _ j$ and $\nu $ are carried by $K_0$,

and similarly for $\nu $, since $\nu (G)\leq \omega (\operatorname {Cap}_ B(G))\leq \omega (\kappa )$. Since $\int g\, \mathrm d\nu _ j\to \int g\, \mathrm d\nu $, we get $\varlimsup _ j|\int f\, \mathrm d\nu _ j-\int f\, \mathrm d\nu |\leq 4C\omega (\kappa )$. Letting $\kappa \downarrow 0$ proves the assertion for the fixed weight.

For the varying weights, let $\delta >0$. Since $|f_ j-f|\leq 2C$ and $\nu _ j$ is carried by $K_0$,

As $j\to \infty $, the capacity on the right tends to zero, and hence so does the second term. Letting $\delta \downarrow 0$ gives $\int |f_ j-f|\, \mathrm d\nu _ j\to 0$, which together with the fixed-weight case proves the first assertion.

For the level sets, fix $s>0$ and let $\sigma ^-_ s,\sigma ^+_ s\colon \mathbb R\to [0,1]$ be the continuous piecewise linear functions such that $\sigma ^-_ s$ equals one on $(-\infty ,c-s]$, zero on $[c,\infty )$ and is linear in between, and $\sigma ^+_ s$ equals one on $(-\infty ,c]$, zero on $[c+s,\infty )$ and is linear in between. Then

Since $\sigma ^\pm _ s$ are Lipschitz, $\sigma ^\pm _ s\circ f_ j$ converge to $\sigma ^\pm _ s\circ f$ in the same sense as $f_ j\to f$, and $\sigma ^\pm _ s\circ f$ is quasi-continuous on a neighborhood of $K_0$ (remark following definition 7.5.1). The first assertion therefore gives

As $s\downarrow 0$, the two outer terms tend to $\nu (\{ f<c\} )$ and $\nu (\{ f\leq c\} )$, which are equal since $\nu (\{ f=c\} )=0$. This proves (7.11.28). $\square$