Conditionally atomless extensions of sigma algebras

TL;DR

This paper introduces two equivalent definitions for sigma algebras that are atomless when conditioned on a smaller sigma algebra, providing a foundational understanding of their structure.

Contribution

It offers a new characterization of conditionally atomless sigma algebras through two equivalent definitions, enhancing theoretical understanding.

Findings

Two equivalent definitions of conditionally atomless sigma algebras

Clarification of their structural properties

Foundation for further theoretical developments

Abstract

We give two equivalent definitions of sigma algebras that are atomless conditionally to a smaller sigma algebra.