On the existence of a probability measure compatible with a total preorder on a Boolean algebra
Let (\({\mathcal A},\lesssim)\) be a totally preordered Boolean algebra of subsets of a give n set \(\Omega\). Then (\({\mathcal A},\lesssim)\) is said to be weakly Archimedean, if for all (A,B)\(\in {\mathcal A}^ 2\), \(A>B\), there exists \(n(A,B)\in N^*\) such that (1): \(kI_{\Omega}-n(I_ A-I_ B)=\sum_{i\in I}(I_{C_ i}-I_{D_ i})\) with \(k\in N\), \(n\in N^*\), \(C_ i\gtrsim D_ i\), \(i\in I\) finite implies (2): \(k/n>1/n(A,B)\). (\({\mathcal A},\lesssim)\) is said to be well bounded, if \(\emptyset <\Omega\) and \(\emptyset \lesssim A\lesssim \Omega\) for all \(A\in {\mathcal A}\). Ultimately, (\({\mathcal A},\lesssim)\) is perfectly separable, if there exists a countable set \({\mathcal C}\), \({\mathcal C}\subset {\mathcal A}\), such that for all (A,B)\(\in {\mathcal A}^ 2\), \(A<B\), there exists \(C\in {\mathcal C}\) such that \(A\lesssim C\lesssim B\). The main result is as follows: There exists a compatible probability measure on (\({\mathcal A},\lesssim)\) if and only if (\({\mathcal A},\lesssim)\) is well bounded, weakly Archimedean, and perfectly separable.
totally preordered Boolean algebra of subsets
scientific article; zbMATH DE number 3920150
scientific article