1,948 research outputs found

    Representation of States on Effect-Tribes and Effect Algebras by Integrals

    Full text link
    We describe Ļƒ\sigma-additive states on effect-tribes by integrals. Effect-tribes are monotone Ļƒ\sigma-complete effect algebras of functions where operations are defined by points. Then we show that every state on an effect algebra is an integral through a Borel regular probability measure. Finally, we show that every Ļƒ\sigma-convex combination of extremal states on a monotone Ļƒ\sigma-complete effect algebra is a Jauch-Piron state.Comment: 20 page

    Smearing of Observables and Spectral Measures on Quantum Structures

    Full text link
    An observable on a quantum structure is any Ļƒ\sigma-homomorphism of quantum structures from the Borel Ļƒ\sigma-algebra of the real line into the quantum structure which is in our case a monotone Ļƒ\sigma-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean Ļƒ\sigma-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure

    Toward a probability theory for product logic: states, integral representation and reasoning

    Full text link
    The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.Comment: 27 pages, 1 figur

    States on pseudo effect algebras and integrals

    Full text link
    We show that every state on an interval pseudo effect algebra EE satisfying some kind of the Riesz Decomposition Properties (RDP) is an integral through a regular Borel probability measure defined on the Borel Ļƒ\sigma-algebra of a Choquet simplex KK. In particular, if EE satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of $K.
    • ā€¦
    corecore