Noise as a Boolean algebra of sigma-fields. III. An old question of Jacob Feldman

TL;DR

This paper investigates the structure of noise-type Boolean algebras, showing that the noise-type completion equals the closure if and only if the algebra is classical, clarifying a fundamental property of such algebras.

Contribution

It proves that the noise-type completion of a Boolean algebra equals its closure precisely when the algebra is classical, answering an old question posed by Jacob Feldman.

Findings

C equals the closure if and only if B is classical

C consists of complemented elements of the closure

Main result characterizes classical noise-type Boolean algebras

Abstract

The noise-type completion C of a noise-type Boolean algebra B is generally not the same as the closure of B. As shown in Part I (Introduction, Theorem 2), C consists of all complemented elements of the closure. It appears that C is the whole closure if and only if B is classical (as defined in Part II, Sect. 1a), which is the main result of this Part III.