Constituting Atoms of a $\sigma$ Algebra via Its Generator

TL;DR

This paper presents a weak sufficient condition under which atoms of a sigma algebra can be determined directly from its generator, simplifying the process of identifying these fundamental elements.

Contribution

It introduces a nearly optimal weak condition that allows atoms to be characterized via the generator, advancing understanding of sigma algebra structure.

Findings

A weak sufficient condition for atom determination is proposed.

The condition is shown to be nearly the weakest possible.

The method simplifies identifying atoms in sigma algebras.

Abstract

To constitute atoms of a $\sigma$ algebra is not a easy task due to the large number of its elements. However, determining them via generators seems a feasible and simple way since most $\sigma$ algebras are generated by their smaller proper subsets. Precisely, under some conditions each atom of a $\sigma$ algebra equals the intersection of the elements containing a point of the atom in the generator. In this paper, a very weak sufficient condition for determining atoms by the generator is presented. The condition, though not being a necessary one, is shown to be almost the weakest one in the sense that it can hardly be improved.