Classical and quantum conditional measures from a categorical viewpoint

TL;DR

This paper develops a categorical framework for classical and quantum measure spaces, extending key theorems and introducing a quantum analogue of conditional measures to unify their treatment.

Contribution

It introduces a categorical approach to classical and quantum measure spaces, extending fundamental theorems and defining quantum conditional measures as duals of classical ones.

Findings

Extended Riesz-Markov-Kakutani theorem to categorical setting

Extended Gelfand duality theorem to an equivalence of categories

Proposed a quantum version of conditional measures as duals of classical measures

Abstract

This paper presents categorical structures on classical measure spaces and quantum measure spaces in order to deal with canonical maps associated with conditional measures as morphisms. We extend the Riesz-Markov-Kakutani representation theorem and the Gelfand duality theorem to an equivalence of categories between them. From this categorical viewpoint, we introduce a quantum version of conditional measures as a dual concept of the classical one.