= Le Cam's first lemma
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= Le Cam first lemma
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{synonym}
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 $\mathbb EL=1$, then $P_n,Q_n$ are <mutually contiguous>. Indeed, $\mathbb E L_n\le1$ 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 $c\downarrow0$.
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