Consider a completely metrizable sapce $\mathcal{X}$ and its borel sigma algebra $\mathcal{F}$. Let $PM$ be the collection of all probability measures defined on $(\mathcal{X}, \mathcal{F})$.
It is known that $\Gamma_1 = \{ Q \in PM: KL(Q ||P) \leq c\}, c > 0$ is compact in $\tau$-topology (the topology of setwise convergence of PM). (see lem 2.3) The key is to show that $P$ uniformly dominates $\Gamma_1$, i.e., for any $\epsilon >0$, there exists $\delta >0$ such that $\forall Q\in \Gamma_1, \forall B\in \mathcal{F}$ , $P(B ) < \delta \implies Q(B) < \epsilon $.
Is the reverse KL ball $\Gamma_2 = \{ Q \in PM: KL(P || Q) \leq c\}, c > 0$ also compact? Is it compact in the topology of weak convergence?
edit: I realize that when Q is not dominated by P, there is no hope that $\Gamma_2$ is compact as Qs can be anything outside $supp(P)$.
So, suppose $P$ has a full support (it dominates all PM), then do we have the compactness for $\Gamma_2$? Or under what condition added to $P$ can yields the compactness?