= Bayes formula for a dominated observation model
{c}
Suppose an unknown has <prior distribution> $\Pi$ and its conditional observation law $P_u$ has jointly measurable <Radon-Nikodym derivative> $L(u,m)$ relative to a fixed observation law $P_0$. If $L>0$ for $\Pi(du)P_0(dm)$-almost every pair, then
$$
Z(m)=\int L(u,m)\Pi(du),\qquad
\frac{d\Pi^m}{d\Pi}(u)=\frac{L(u,m)}{Z(m)}
$$
defines the <posterior distribution> for almost every observation. Indeed, $\int L(u,m)P_0(dm)=1$ for almost every $u$, so <Tonelli theorem> gives $\int Z(m)P_0(dm)=1$. Hence $Z$ is finite almost everywhere, and positivity of $L$ and <Fubini's theorem> give $Z>0$ almost everywhere. The data law is $ZP_0$, and integration of the proposed posterior against this data law recovers the joint law. This proves the <conditional distribution> property and also shows that the data law is an <equivalent probability measure> to $P_0$.
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