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

TL;DR

This paper characterizes $\sigma$-additive states on effect-tribes using integrals and demonstrates that all states can be represented via Borel measures, with implications for extremal states and convex combinations.

Contribution

It provides a novel integral representation of states on effect-tribes and establishes that all states are integrals over Borel measures, extending the understanding of effect algebra states.

Findings

States on effect-tribes are represented by integrals with Borel measures

Every state on an effect algebra can be expressed as an integral over a Borel measure

Convex combinations of extremal states are Jauch-Piron states

Abstract

We describe $\sigma$-additive states on effect-tribes by integrals. Effect-tribes are monotone $\sigma$-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 $\sigma$-convex combination of extremal states on a monotone $\sigma$-complete effect algebra is a Jauch-Piron state.