Proposition 23.1.47316CC905C469

Proof. For bounded $u\leq 0$, apply lemma 23.1.46. The bounded capacity continuity of products and quasi-continuous coefficients in section 5.27, section 5.29 passes the total estimate (23.1.45) to $u$. For the tail at $T>0$, use continuous depth cutoffs majorizing $\mathbf1_{\{ -u>T\} }$ and supported in $\{ -u>T-\delta \} $, where $0<\delta <T$. The same product continuity gives a bound by $12Q_{T-\delta }(W)$. Continuity of $(1+\sqrt{2t})W(t)$ gives $Q_{T-\delta }(W)\to Q_ T(W)$. For $T=0$, use the total estimate.

For unbounded $u$, the canonical truncations $u^ K=\max (u,-K)$ remain bounded and invariant. On $\{ u>-K\} $ their measures agree with the finite-level restrictions defining $\mu _ u$. These restrictions increase to $\mu _ u$, so monotone convergence proves (23.1.48).

Splitting at depth $T$ and rescaling $u^ T$ into a capacity test gives, for every Borel set $B$,

Indeed $\omega +\mathrm{dd}^{\mathrm{c}}u^ T\leq (1+T)(\omega +\mathrm{dd}^{\mathrm{c}}(u^ T/(1+T)))$, and the latter potential has range of length at most one. If $\sqrt t\, W(t)\to 0$, then $Q_ T(W)\to 0$. Taking first $T$ large and then the capacity small proves uniform absolute continuity.

Suppose $u_ j\to u$ in capacity in $\mathcal R$. For fixed $K$, the truncations $u_ j^ K$ converge in capacity to $u^ K$, and their weighted bounded products converge weakly. Outside $\{ u\leq -K\} $, locality identifies $\nu _{W,u^ K}$ with $\nu _{W,u}$. Both measures give this exceptional set mass at most $12Q_{K-1}(W)$ when $K>1$, by their tail estimates; this also includes any truncation mass on $\{ u=-K\} $. Hence

uniformly over $u\in \mathcal R$. Letting $K\to +\infty $ removes the truncations and proves the claimed continuity. $\square$