Decompositions of Measures on Pseudo Effect Algebras

TL;DR

This paper explores measure decompositions on pseudo effect algebras with the Riesz Decomposition Property, establishing existence and uniqueness results by leveraging the structure of their state spaces.

Contribution

It introduces measure decomposition theorems for pseudo effect algebras satisfying (RDP), extending classical measure theory to this algebraic context.

Findings

Existence of measure decompositions analogous to Yosida-Hewitt and Lebesgue types.

Uniqueness of these decompositions due to the simplex structure of the state space.

Characterization of the state space as either empty or a nonempty simplex.

Abstract

Recently in \cite{Dvu3} it was shown that if a pseudo effect algebra satisfies a kind of the Riesz Decomposition Property ((RDP) for short), then its state space is either empty or a nonempty simplex. This will allow us to prove a Yosida-Hewitt type and a Lebesgue type decomposition for measures on pseudo effect algebra with (RDP). The simplex structure of the state space will entail not only the existence of such a decomposition but also its uniqueness.