Likelihood-ratio classes are minimal sufficient on a common discrete support

Fix a class $D\subseteq X_0$ and its first point $r_D$ in a fixed enumeration. By [F1], $w_D(x)=P_{\theta }(\{x\})\mathrm{/}{P}_{\theta }(\{r_D\})$ for $x\in D$ is positive and independent of $\theta$. The sum $W_D={\sum }_{x\in D}{w}_D(x)=P_{\theta }(D)\mathrm{/}{P}_{\theta }(\{r_D\})$ is finite and positive. Define $K(D,A)={\sum }_{x\in A\cap D}{w}_D(x)\mathrm{/}{W}_D$ for every $A\subseteq \mathcal{X}$. Each row is a probability measure: its total mass is one and countable additivity follows by regrouping nonnegative sums over disjoint events. Every row is independent of $\theta$, and the countable discrete quotient makes all evaluations measurable [F4]. This defines $K$ on every class; off-support sample points use the row of their assigned class, with no separate row assignment.

Let $S$ be any other sufficient statistic into a standard Borel space, with common kernel $L$. If $x,y\in X_0$ and $S(x)=S(y)=s$, then $\{s\}$ is Borel and $P_{\theta }(S=s)>0$ for every $\theta$. The conditional identity forces $L(s,\{x\})=P_{\theta }(\{x\})\mathrm{/}{P}_{\theta }(S=s)$ and $L(s,\{y\})=P_{\theta }(\{y\})\mathrm{/}{P}_{\theta }(S=s)$. The common kernel has no parameter index, so their ratio is constant in $\theta$; [F1] yields $x\sim y$. Therefore $S$ is constant only within one class on $X_0$.

For every $\theta$, class $D$, and $A\subseteq \mathcal{X}$, the point-mass ratio definition gives $P_{\theta }(A\cap D)=P_{\theta }(D)K(D,A)$. Summing over classes proves the conditional-law identity in [F2]; thus $T$ is sufficient.

Since $X_0$ is countable, its image under $S$ is countable and Borel in the standard Borel target. Step 1.2 defines a function from that image to the countable quotient with $T=\phi \circ S$ on $X_0$; extend $\phi$ by one fixed class off the image. This extension is Borel because its nonconstant domain is countable. Since every model law is concentrated on $X_0$, $T=\phi (S)$ almost surely under every parameter. This proves the comparison with every standard Borel sufficient statistic.

Let $\mathcal{H}$ be any sufficient sub-sigma-field of the full sigma-field on $\mathcal{X}$. For $x,y\in \mathcal{X}$ that some $\mathcal{H}$-event separates, choose one such event, using [A1] for this countable set of pairs. The atoms $H_x=\{y:\text{no }\mathcal{H}\text{-event separates }x,y\}$ are measurable: each is the countable intersection of the chosen events or their complements containing $x$. Every $\mathcal{H}$-measurable function is constant on each $H_x$, since unequal real values would be separated by the inverse image of a rational cut. For $x,y\in X_0$ in the same atom $H$, let $v_x,v_y$ be common bounded conditional-expectation versions of $1_{\{x\}},1_{\{y\}}$ supplied by sufficiency. Testing on $H$ gives $v_x{\mathrm{\mid }}_H=P_{\theta }(\{x\})\mathrm{/}{P}_{\theta }(H)$ and $v_y{\mathrm{\mid }}_H=P_{\theta }(\{y\})\mathrm{/}{P}_{\theta }(H)$ for every $\theta$; these constants are positive and independent of $\theta$. Thus the point-mass ratio is constant and $x\sim y$. Each class $D$ is therefore the intersection with $X_0$ of a union of $\mathcal{H}$-atoms. That union is countable and measurable, and differs from $T^{-1}(\{D\})$ only off $X_0$. Unions of classes give $\sigma (T)\subseteq \mathcal{H}$ modulo the family-null ideal. The common kernel in step 2.1 supplies bounded conditional versions for $\sigma (T)$ by summing bounded functions on the countable sample space. Hence [F3] proves minimal sufficiency in its full sigma-field sense.