Combinations of Observables

Abstract

This article begins with a review of the framework of fuzzy probability theory.The basic structure is given by the σ-effect algebra of effects (fuzzy events) E(Ω,A) and the set of probability measures M ⁺ ₁ (Ω, A) on a measurable space (Ω,A). An observable X: B → E(Ω, A) is defined, where (Λ, B) is the value spaceof X. It is noted that there exists a one-to-one correspondence between states onE(Ω, A) and elements of M ⁺ (Ω, A) and between observables X: B → E(Ω,A) and σ-morphisms from E(Λ, B) to E(Ω, A). Various combinations ofobservables are discussed. These include compositions, products, direct products,and mixtures. Fuzzy stochastic processes are introduced and an application toquantum dynamics is considered. Quantum effects are characterized from amonga more general class of effects. An alternative definition of a statistical map T:M ⁺ ₁ (Ω, A) → M ⁺ ₁ (Λ, B) is given.