Corollary 11.39.6FCFB63397013

Corollary 11.39.6.FCFB63397013 Let the model references $\varphi _ j$ satisfy $P_{\varepsilon _ j\theta }(\varphi _ j)\not\equiv -\infty $, where $0<\varepsilon _ j<1$ and $\varepsilon _ j\to 0$. Then $B_{\varphi _ j}\to B_0$ uniformly on $C(X)$. Let $G_ j,G\in C(X)$ with $G_ j\to G$ uniformly, and let $\nu _ j\in \mathcal A_{\varphi _ j}(G_ j)$. Then every weak cluster point of $(\nu _ j)$ belongs to $\mathcal A_0(G)$, the densities of the $\nu _ j$ are uniformly integrable, and a weakly convergent subsequence also converges weakly in $L^1(\mathrm{d}V)$. If $\mathcal A_0(G)$ is a singleton, the whole sequence converges to it in both senses.