Measures on circle coarse-grained systems of sets
Let \(\mathcal{C}\) be the unit circle parametrized within \([0,2\pi[\), \(n\in\mathbb{N}\) and \(\Delta_n\) the quantum logic generated by the intervals \([a,a+\frac{2\pi}{n}[\), \(a\in [0,2\pi[\), i.e. the smallest system of subsets of \(\mathcal{C}\) which contains these intervals and is closed under the formation of disjoint unions and complements in \(\mathcal{C}\). The main theorem says that any finitely additive probability measure \(m:\Delta_n\rightarrow [0,1]\) can be extended to a measure over the Boolean algebra of all subsets of \(\mathcal{C}\). As observed by the authors standard techniques cannot be used for the proof since \(\Delta_n\) is not closed under intersections.
scientific article; zbMATH DE number 5770260