Let $\Delta_{\mathcal A}$ be the set of all finitely additive probability measures on a $\sigma$-field $\mathcal A$. The so-called "natural" $\sigma$-field $\mathcal D$ on $\Delta_{\mathcal A}$ is the one generated by the mappings $P \mapsto P(A)$, $A \in \mathcal A$. Or, in other words, $\mathcal D$ is generated by sets of the form $\{P \in \Delta_{\mathcal A}: P(A) \in B\}$, where $A \in \mathcal A$ and $B$ is a Borel subset of $\mathbb R$.
Is the set $\Delta_{\mathcal A}^{ca}$ of all countably additive probability measures on $\mathcal A$ measurable? That is, is $\Delta_{\mathcal A}^{ca} \in \mathcal D$?