Observables on lexicographic MV-algebras

Abstract

We study observables on a special kind of MV-algebras which are lexicographic MV-algebras, i.e. they are isomorphic to MV-algebras of the form M=Γ(H×→G,(u,0)), where (H,u) is a unital subgroup of (R,1) and G is a Dedekind σ-complete ℓ-group. Observables are a special kind of σ-homomorphisms defined on the Borel σ-algebra of the real line with values in the lexicographic MV-algebra. They model measurable functions. Spectral resolution is a system of elements of the MV-algebra indexed by real numbers which is monotone, “left continuous”, and going to 0 and 1 if t goes to −∞ and to +∞, respectively. The main task of the paper is to show when does a spectral resolution {xt:t∈R} determine an observable x such that x((−∞,t))=xt for each t∈R.