Standard Borel dominated factorization criterion

By [F4], choose a positive-weight countable mixture $Q={\sum }_j{w}_j{P}_{{\theta }_j}$ dominating the family, and let $h=dQ\mathrm{/}d\mu$. This RN derivative exists under AC because $Q$ is a probability and $Q\ll \mu$.

Suppose $T$ is sufficient. Let $K$ be the common conditional law from [F1]. For each $A,C$, sum the conditional-law identity under $P_{{\theta }_j}$ with weights $w_j$; [F9] applied to increasing partial sums gives the same identity under $Q$. Thus the same $K$ is common for the augmented family $\{Q\}\cup \{P_{\theta }\}$.

Conversely suppose the factorization holds. By [F4], $P_{\theta }\ll Q$ for every $\theta$. The finite measures $P_{\theta },Q$ and sigma-finite $\mu$ have a common finite exhaustion, so the RN chain rule [F6] identifies $d{P}_{\theta }\mathrm{/}dQ=p_{\theta }\mathrm{/}h$ on the $Q$-full set $\{h>0\}$. There the assumed equality gives $d{P}_{\theta }\mathrm{/}dQ=g_{\theta }(T)$.

In this augmented family $Q$ is a member and dominates all laws. Apply [F5] to obtain Borel $g_{\theta }$ with $d{P}_{\theta }\mathrm{/}dQ=g_{\theta }(T)$ $Q$-almost surely. A sigma-finite cover for $\mu$ is a common finite-measure exhaustion for $P_{\theta },Q,\mu$, since the first two are finite; the chain rule [F6] and density identities [F3, F7] then give $p_{\theta }=h{g}_{\theta }(T)$ $\mu$-almost everywhere. The exceptional set may depend on the fixed $\theta$.

Also $dQ\mathrm{/}dQ=1$ is a function of $T$. In the augmented family $\{Q\}\cup \{P_{\theta }\}$, $Q$ is a member reference and all laws are dominated by it; apply [F5] to these T-measurable derivatives. The criterion gives a common conditional law, so $T$ is sufficient.