A countable mixture likelihood-ratio vector is minimal sufficient

Since $P_k\ll Q$, [F5] gives finite-valued nonnegative Borel densities $f_k$ after changing each on its null set. Their exceptional sets have countable union $N$ with $Q(N)=0$; reset all $f_k$ to zero on $N$. By [F6], $[0,\mathrm{\infty }{)}^{\mathbb{N}}$ is standard Borel. The coordinate maps are Borel, hence $T$ is Borel.

By [F7], choose a conditional-law kernel $K$ for $X$ given $T$ under $Q$. Its defining identity extends from indicator weights to nonnegative Borel weights by [F9]. By [F5], $P_k(A\cap T^{-1}C)=\int 1_A{1}_C(T)f_{k}dQ$. Since $T_k=f_k$, applying the weighted conditional identity with weight $1_C(t)t_k$ gives $P_k(A\cap T^{-1}C)={\int }_{C}K(t,A)t_k{Q}^T(dt)$; taking $A=\mathcal{X}$ gives $P_k^T(C)={\int }_C{t}_k{Q}^T(dt)$. Thus $K$ works under every $P_k$, so [F1] gives sufficiency.

Let $S$ be any other standard Borel sufficient statistic with common kernel $L$. The same kernel is a conditional law under $Q$: its conditional identity follows by multiplying the identity under $P_k$ by $w_k$ and summing the nonnegative integrals, with [F9] justifying passage to the countable sum. It is also the common conditional law under each $P_k$. Let $Q^S$ and $P_k^S$ be the image laws. The identity $P_k^S\ll Q^S$ follows from $P_k\ll Q$; write $r_k=d{P}_k^S\mathrm{/}d{Q}^S$. Disintegrating with $L$ and extending the conditional identity to the nonnegative weight $r_k$ by [F9] gives, for every $A$, $P_k(A)=\int L(s,A)r_k(s)Q^S(ds)={\int }_A{r}_k(S(x))Q(dx)$. RN uniqueness yields $f_k=r_k(S)$ $Q$-almost surely.

Apply [F5] to each coordinate to obtain Borel $r_k$ (modified on $Q^S$-null sets if necessary). The map $\phi (s)=(r_k(s){)}_{k\ge 1}$ is Borel because its coordinate maps are Borel; step 2.2 gives $T=\phi (S)$ $Q$-almost surely. Since every weight is positive, $Q(C)=0$ iff $P_k(C)=0$ for every $k$. For every Borel $D$ in the target of $T$, the sets $T^{-1}(D)$ and $S^{-1}({\phi }^{-1}(D))$ differ only on $\{T\mathrm{\ne }\phi (S)\}$, which is $Q$-null by step 2.2. Therefore $\sigma (T)\subseteq \sigma (S)$ modulo the family-null ideal, establishing the asserted comparison with $S$.

Let $\mathcal{H}$ be any sufficient sub-sigma-field as in [F2]. For each $k$, apply [F5] on $(\mathcal{X},\mathcal{H})$ to obtain a finite nonnegative $\mathcal{H}$-measurable density $r_k=d(P_k{\mathrm{\mid }}_{\mathcal{H}})\mathrm{/}d(Q{\mathrm{\mid }}_{\mathcal{H}})$. For each $A\in \mathcal{A}$, sufficiency supplies one bounded $\mathcal{H}$-measurable $v_A$ with the conditional identities for $1_A$ under every $P_k$. Summing with weights gives those identities under $Q$. Testing on $\{v_A<0\}$ and $\{v_A>1\}$ shows $0\le v_A\le 1$ $Q$-almost surely; clipping on these $\mathcal{H}$-measurable null sets preserves every identity because $P_k\ll Q$. Simple approximation and [F9] extend the $Q$ conditional identity to the nonnegative $\mathcal{H}$-measurable weight $r_k$, while the restricted RN identity extends to $v_A$. Consequently ${\int }_A{r}_{k}dQ=\int v_A{r}_{k}dQ=\int v_{A}d{P}_k=P_k(A)$. Uniqueness in [F5] gives $f_k=r_k$ $Q$-almost surely. Combining only these countably many null sets, $T=(r_k{)}_k$ $Q$-almost surely. The right side is $\mathcal{H}$-measurable, so every event in $\sigma (T)$ agrees modulo a family-null set with an event in $\mathcal{H}$. The kernel in step 2.1 gives sufficiency of $\sigma (T)$ by extending its event identities to bounded functions using simple approximation and [F9]. Thus [F2] proves minimality against every sufficient sub-sigma-field.