PomeProposition 11.34.3.CB2DC1F2B712 Let $F_ j,F\in C(X)$ with $F_ j\to F$ uniformly, and let $\nu _ j\in \mathcal A_\varphi (F_ j)$. Then every weak cluster point of $(\nu _ j)$ belongs to $\mathcal A_\varphi (F)$. In particular, if the latter set consists of a single measure, $\nu _ j$ converges weakly to it.
Proposition 11.34.3CB2DC1F2B712
Proof. Take a weakly convergent subsequence and fix $G\in C(X)$. In the inequality
\[ B_\varphi (G)\leq B_\varphi (F_ j)-\int _ X(G-F_ j)\, \mathrm{d}\nu _ j, \]
the Lipschitz continuity of $B_\varphi $, the uniform convergence of $F_ j$ and the weak convergence of the probabilities allow passage to the limit. The limiting inequality holds for every $G$, so the limit measure belongs to $\mathcal A_\varphi (F)$. If that set is a singleton, every convergent subsequence has the same limit, and compactness of the space of probability measures proves the final assertion. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.