A base-null observation is a prior-predictive null fiber

The nonempty two-point discrete space $\mathsf{X}$ is standard Borel by [F6], and $\alpha ={\delta }_a$ is a probability measure by [F6], hence a finite nonzero base measure of total mass $1$. Thus [F1] applies: $G(\{b\})=0$ almost surely. Since $\mathrm{Pr}(X_1=b\mid G)=G(\{b\})$, integration over the prior gives $\mathrm{Pr}(X_1=b)=0$, also agreeing with [F3].

The formula in [F2] assigns $K(b)=DP({\delta }_a+{\delta }_b)$ at the null data value. Define another kernel by $K^{\mathrm{\prime }}(a)=K(a)$ and $K^{\mathrm{\prime }}(b)=DP({\delta }_a)$. Both sections are probability measures, and $K^{\mathrm{\prime }}$ is measurable because $\{b\}$ is measurable in the discrete data space. For any measurable event $C$ in $\mathcal{P}(\mathsf{X})$, the disintegration integrals for $K$ and $K^{\mathrm{\prime }}$ differ only at $b$, which has data-marginal mass zero by step 1.1. Thus both are posterior versions; the update formula at $b$ is a version choice, not evidence-supported learning.

In the separate diffuse-base model, [F3] gives data-marginal mass $M(\{y\})=\gamma (\{y\})\mathrm{/}\gamma (\mathsf{Y})=0$. Let $K(z)=DP(\gamma +{\delta }_z)$ be the posterior kernel from [F2] and set $K^{\mathrm{\prime }}(z)=K(z)$ for $z\mathrm{\ne }y$, $K^{\mathrm{\prime }}(y)=DP(\gamma )$. By [F4], $\{y\}$ is measurable. For every measurable $C\subseteq \mathcal{P}(\mathsf{Y})$, $z\mapsto K^{\mathrm{\prime }}(z,C)={\mathbf{1}}_{\mathsf{Y}\setminus \{y\}}(z)K(z,C)+{\mathbf{1}}_{\{y\}}(z)DP(\gamma )(C)$ is measurable; each section is a probability measure, so [F5] makes $K^{\mathrm{\prime }}$ a probability kernel. The disintegration integrals for $K$ and $K^{\mathrm{\prime }}$ agree on every measurable rectangle because the kernels differ only on $\{y\}$ and $M(\{y\})=0$. Thus $K^{\mathrm{\prime }}$ is also a posterior version, proving that the displayed update value at this fixed null fiber is not uniquely determined.