 Suppose we are given a statistical model \((\mathcal{A},\Delta)\). A very straightforward approach to constructing a “classical interpretation” of \((\mathcal{A},\Delta)\) would begin by trying to embed \(\mathcal{A}\) in a Borel test space \(\mathcal{B}\), with the hope of then accounting for the statistical states in \(\delta\) as averages over “hidden” classical—that is, dispersion-free—states on the latter. Thus, we’d want to find a set \(S\) and a mapping \(X \rightarrow \wp(S)\) assigning to each outcome \(x\) of \(\mathcal{A}\) a set \(x* \subseteq S\) in such a way that, for each test \(E \in \mathcal{A}, \{x* \mid x \in E\}\) forms a partition of \(S\). If this can be done, then each outcome \(x\) of \(\mathcal{A}\) simply records the fact that the system is in one of a certain set of states, namely, \(x\)*. If we let \(\Sigma\) be the \(\sigma\)-algebra of sets generated by sets of the form \(\{x* \mid x \in X\}\), we find that each probability measure \(\mu\) on \(\Sigma\) pulls back to a state \(\mu\)* on \(\mathcal{A}\), namely, \(\mu *(x) = \mu(x\)*). So long as every state in \(\delta\) is of this form, we may claim to have given a completely classical interpretation of the model \((\mathcal{A},\Delta)\).