Noise as a Boolean algebra of sigma-fields. I. Completion

TL;DR

This paper develops a new framework to extend nonclassical planar noises, like black noise of percolation, by treating less regular domains as limits of regular ones, thereby broadening the set of sigma-fields.

Contribution

It introduces a novel general framework for extending planar noises through limits of regular domains, enhancing the understanding of nonclassical noises.

Findings

Extended the set of sigma-fields for planar noises.

Provided a new approach to treat less regular domains as limits.

Established a comprehensive framework for noise extension.

Abstract

Nonclassical noises over the plane (such as the black noise of percolation) consist of sigma-fields corresponding to some planar domains. One can treat less regular domains as limits of more regular domains, thus extending the noise and its set of sigma-fields. The greatest extension is investigated in a new general framework.