Lemma 16.28.33A143B4417F78

Proof. Nonemptiness follows from theorem 16.28.27. Since the test functions are continuous, $\mathcal J_ a(K)$ is closed and convex, and the equality defining active pairs is a closed affine condition. For an active pair, (16.28.17), applied with the weight $q$, gives

The pairs in $\mathcal J_ a(K)$ with bounded constants form a compact set, as in the proof of theorem 16.28.29; hence the set of active pairs is compact.

With $q$ fixed, lemma 16.28.18 and (16.28.17), applied with the weight $q$, bound $|F_{\eta ,q}(a)-F_{\delta ,q}(a)|$ by $|\eta -\delta |(V_ K^*(a)+\operatorname {osc}_ Kq/\epsilon )$. Comparison of obstacles and translation by constants bound $|F_{\eta ,p}(a)-F_{\eta ,q}(a)|$ by $\| p-q\| _\infty $. Adding these bounds gives (16.28.34).

For the sequence in the statement, the constants $c_ j$ have a common bound, by the estimate above, because the weights $q_ j$ are uniformly bounded and the parameters $\delta _ j$ are bounded below by a positive constant. Weak compactness of probability measures gives a convergent subsequence of pairs, and its limit belongs to $\mathcal J_ a(K)$, which is closed. The estimate (16.28.34) gives $F_{\delta _ j,q_ j}(a)\to F_{\delta ,q}(a)$, while uniform convergence of the $q_ j$ and weak convergence of the $\mu _ j$ give $\int _ Kq_ j\, \mathrm{d}\mu _ j\to \int _ Kq\, \mathrm{d}\mu $. Passing to the limit in the equality defining active pairs proves the last assertion. $\square$