Every State on Interval Effect Algebra is Integral

TL;DR

This paper proves that all states on interval effect algebras can be represented as integrals over probability measures, extending to subalgebras and effect operators in Hilbert spaces, unifying various algebraic structures.

Contribution

It establishes a general integral representation for states on interval effect algebras, including MV-algebras and effect operators in Hilbert spaces, and shows extension properties for subalgebras.

Findings

States on interval effect algebras are representable as integrals over Borel probability measures.

States on effect subalgebras can be extended to the entire algebra.

Representation applies to effect operators in Hilbert spaces.

Abstract

We show that every state on an interval effect algebra is an integral through some regular Borel probability measure defined on the Borel $\sigma$-algebra of a compact Hausdorff simplex. This is true for every effect algebra satisfying (RDP) or for every MV-algebra. In addition, we show that each state on an effect subalgebra of an interval effect algebra $E$ can be extended to a state on $E.$ Our method represents also every state on the set of effect operators of a Hilbert space as an integral