States on pseudo effect algebras and integrals

TL;DR

This paper demonstrates that states on certain pseudo effect algebras can be represented as integrals over Borel probability measures on a Choquet simplex, with uniqueness under stronger conditions.

Contribution

It establishes a representation theorem linking states on pseudo effect algebras satisfying Riesz Decomposition Properties to integrals over Choquet simplices, including uniqueness results.

Findings

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

Unique measure representation when the strongest RDP holds.

Connection between algebraic properties and measure-theoretic representations.

Abstract

We show that every state on an interval pseudo effect algebra $E$ satisfying some kind of the Riesz Decomposition Properties (RDP) is an integral through a regular Borel probability measure defined on the Borel $\sigma$-algebra of a Choquet simplex $K$. In particular, if $E$ 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 $K.$