Finite function spaces and measures on hypergraphs (Q1115883)

A measure on a hypergraph \(H=(X,{\mathcal O})\) is a real valued function on X whose sum over any set in \({\mathcal O}\) is constant, the strength of the measure. Various types of measures have been considered in the literature. States or stochastic models are non-negative measures of strength one. These have been studied by graph theorists as generalizations of doubly stochastic matrices. In operational statistics they are viewed as probability measures. In quantum logic theory, the elements of X correspond to elementary experimental propositions and a state gives the probability that a proposition is true. The paper begins with a study of finite linear function spaces. These spaces have an independent interest. The results so obtained are then applied to measures on hypergraphs. In particular various measure morphisms and isomorphisms are characterized. Furthermore, characterizations and sufficient conditions are given for the set of Jordan measures to be equal to the total set of measures. Finally, the dimension of the space of measures on a hypergraph is investigated.