 Le Cam's first lemma

Let $L_n$ be the probability density function of the absolutely continuous part of $Q_n$ relative to $P_n$. If $L_n$ converges in distribution under $P_n$ to $L>0$ almost surely with $\mathbb{E}L=1$, then $P_n,Q_n$ are mutually contiguous. Indeed, $\mathbb{E}{L}_n\le 1$ and convergence to a mean-one limit imply uniform integrability and vanishing singular mass, giving forward contiguity. For reverse contiguity, $P_n(A_n)\le P_n(L_n\le c)+Q_n(A_n)/c$, and one first takes $n\to \infty$ and then $c\downarrow 0$.