Likelihood-ratio equivalence on common discrete support

Let $\mathcal{X}$ be countable with its full $\sigma$-field, so it is standard Borel under the discrete metric. Let $\mathrm{\Theta }\mathrm{\ne }\mathrm{\varnothing }$, and let $(P_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ be dominated by a fixed $\sigma$-finite positive measure $\mu$. Choose the finite-valued densities $p_{\theta }=d{P}_{\theta }\mathrm{/}d\mu$ as in Likelihood densities. Let $X_0\subseteq \mathcal{X}$ satisfy $P_{\theta }(X_0)=1$ and $P_{\theta }(\{x\})>0$ for every $x\in X_0$ and every $\theta$. Write $m_{\theta }(x)=P_{\theta }(\{x\})$. For each $x\in X_0$, $0<\mu (\{x\})<\mathrm{\infty }$ and $p_{\theta }(x)=m_{\theta }(x)\mathrm{/}\mu (\{x\})$: positivity follows from $m_{\theta }(x)>0$, finiteness from sigma-finiteness, and the value is fixed by the density integral on the positive-measure singleton. Consequently

$\frac{p_{\theta }(x)}{p_{\theta }(y)}=\frac{m_{\theta }(x)}{m_{\theta }(y)}\frac{\mu (\{y\})}{\mu (\{x\})}\mathrm{.}$

For $x,y\in X_0$, define $x\sim y$ if there is a constant $c(x,y)>0$ such that $m_{\theta }(x)\mathrm{/}{m}_{\theta }(y)=c(x,y)$ for every $\theta$. The displayed fixed singleton factor shows this is equivalent to constancy of the common-reference density ratio. This is an equivalence relation: the constants for $(x,x)$, $(y,x)$ and $(x,z)$ are respectively $1$, $c(x,y{)}^{-1}$ and $c(x,y)c(y,z)$. Its equivalence classes form a countable quotient. Equip that quotient with its discrete $\sigma$-field; it is standard Borel because the discrete metric is complete and the countable space itself is dense. Since $\mathrm{\Theta }\mathrm{\ne }\mathrm{\varnothing }$ and $P_{\theta }(X_0)=1$, $X_0$ is nonempty; define the class statistic by mapping each point of $X_0$ to its class and every point off $X_0$ to any fixed class. It is measurable because $\mathcal{X}$ has the full $\sigma$-field, and values off $X_0$ are null under every law.