Theorem 13.28.83F90EA1D636A

Proof. Write $\mu _ j=\mu _{p_ j}$ and $\mu =\mu _ p$. By (13.28.5), $\mu _ j\to \mu $ in total variation. Thus $U(p_ j)=A-\mu _ j(Y)\to U(p)$. If $U(p)>0$, the models lie in a common $\mathcal M_\delta (\theta )$ for all large $j$, and proposition 13.22.4, theorem 13.23.3 apply.

Suppose that $U(p)=0$. Then $T_ p=\mu $ and $\mu (Y)=A$, while $T_{p_ j}-\mu _ j$ is a positive measure of mass $A-\mu _ j(Y)=\mu (Y)-\mu _ j(Y)$. Consequently

which proves (13.28.9). By normalized compactness, each subsequence has a further subsequence converging in $L^1$ to a function $q$ with supremum zero. Then $T_{p_ j}\to T_ q$ weakly, so $T_ q=T_ p$ and $\mathrm{dd}^{\mathrm{c}}(q-p)=0$. Hence $q-p$ is constant, and the normalization gives $q=p$. This proves $L^1$ convergence of the whole sequence. Finally, apply lemma 5.27.11 on finitely many nested coordinate disks: both terms on its right-hand side tend to zero, and the comparison between local and global capacities in section 5.29 gives convergence in capacity. $\square$