Noise as a Boolean algebra of $\sigma$-fields

TL;DR

This paper investigates the structure of noises as homomorphisms from Boolean algebras to $\sigma$-fields, revealing that the maximal extension is complete iff the noise is classical, thus answering a longstanding question.

Contribution

It characterizes when the Boolean algebra of $\sigma$-fields is complete, linking this property to the classicality of the noise, and resolves an open problem by J. Feldman.

Findings

Maximal extension of Boolean algebra of $\sigma$-fields is complete iff the noise is classical.

Provides a new characterization of classical noise in terms of Boolean algebra properties.

Answers an open question posed by J. Feldman.

Abstract

A noise is a kind of homomorphism from a Boolean algebra of domains to the lattice of $\sigma$-fields. Leaving aside the homomorphism we examine its image, a Boolean algebra of $\sigma$-fields. The largest extension of such Boolean algebra of $\sigma$-fields, being well-defined always, is a complete Boolean algebra if and only if the noise is classical, which answers an old question of J. Feldman.