Likelihood-tilted measures at a KL projection

Fix a selected ${\theta }^{\ast }\in {\mathrm{\Theta }}^{\ast }$ as in KL projection sets and relative KL neighborhoods. Since its KL value is finite, $p_{{\theta }^{\ast }}>0$ $P_0$-almost surely. For each $\theta \in \mathrm{\Theta }$, define the nonnegative set function $Q_{\theta }(A):={\int }_A\frac{p_{\theta }(x)}{p_{{\theta }^{\ast }}(x)}{P}_0(dx),A\in \mathcal{A},$ where the ratio is assigned value $0$ on the $P_0$-null set $\{p_{{\theta }^{\ast }}=0\}$. Write $q_{\theta }:=Q_{\theta }(\mathsf{X})={\mathbb{E}}_{P_0}\left[\frac{p_{\theta }(X)}{p_{{\theta }^{\ast }}(X)}\right]\in [0,+\mathrm{\infty }]\mathrm{.}$ When $q_{\theta }\mathrm{\ne }1$, $Q_{\theta }$ is not a probability law. The local sigma-finiteness lemma proves that it is a measure. The local product-measure lemma then constructs $Q_{\theta }^n$ and proves its density relative to $P_0^n$ is ${\prod }_{i=1}^n{p}_{\theta }(X_i)\mathrm{/}{p}_{{\theta }^{\ast }}(X_i)$.

For a measurable test ${\varphi }_n:{\mathsf{X}}^n\to [0,1]$, once that product measure has been constructed, $Q_{\theta }^n(1-{\varphi }_n)$ means the nonnegative integral of $1-{\varphi }_n$ under it. It may be infinite; a hypothesis that this quantity tends to zero in particular requires it to be finite for all sufficiently large $n$.