Abstract
Non-commutative measure theory embraces measure theory on cr-fields of subsets of a set, on projection lattices of von Neumann algebras or JBW-algebras and on hypergraphs alike [20], [27], [33], [37], [39], [40], [41]. Due to the unifying structure of an orthoalgebra concepts can easily be transferred from one branch to the other. Additional conceptual inpetus is obtained from the logico-probabilistic foundations of quantum mechanics (see [6], [19], [21]).In the late seventies the author studied the Jordan-Hahn decomposition of measures on orthomodular posets and certain graphs. These investigations revealed an interesting geometrical aspect of this decomposition in that the Jordan-Hahn property of the convex set of probability charges on a finite orthomodular poset can be characterized in terms of the extreme points of the unit ball of the Banach space dual of the base normed space of Jordan charges.
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