 Let \(S\) be a set, understood for the moment as the state-space of a physical system, and let \(\Sigma\) be a \(\sigma\)-algebra of subsets of \(S\). We can regard each partition \(E\) of \(S\) into countably many pair-wise disjoint \(\Sigma\)-measurable subsets as representing a “coarse-grained” approximation to an imagined perfect experiment that would reveal the state of the system. Let \(\mathcal{A}_{\Sigma}\) be the test space consisting of all such partitions. Note that the outcome set for \(\mathcal{A}_{\Sigma}\) is the set \(X = \Sigma \setminus \{\varnothing \}\) of non-empty \(\Sigma\)-measurable subsets of \(S\). Evidently, the probability weights on \(\mathcal{A}_{\Sigma}\) correspond exactly to the countably additive probability measures on \(\Sigma\). 