Consider two probability measures $P$ and $P_0$ having the same support $X \subset \mathbb{R}^d$ and densities $p$ and $p_0$ with respect to the Lebesgue density. Let $X_1, \ldots,X_n$ be iid according to $P_0$ and define the likelihood ratio $\Lambda_n:=\prod_{i=1}^np(X_i)/p_0(X_i)$. The sequence $(\Lambda_n)$ is uniformly tight (e.g. van der Vaart, Asymptotic statistics, 2000, page 88). Moreover, under various conditions involving distances or divergences between $p$ and $p_0$, it can be shown that $\Lambda_n \leq e^{nc}$ eventually $P_0^{(\infty)}$ almost surely, for a positive constant $c$.
Can we then conclude that, for large enough $n$, $\mathbb{E}\Lambda_n \epsilon_{B_n} \leq e^{nc} P_0^{(n)}(B_n)$, for any sequence of measurable sets $B_n$, where $\epsilon_{B_n}$ is the indicator function of the set $B_n$ at $(X_1,\ldots,X_n)$?