A note on the "conditional triviality" property for regular conditional distributions

TL;DR

This paper explores the conditions under which regular conditional distributions trivialize the conditioning sigma-algebra, linking this property to the existence of measurable selections for almost all measures derived from the original distribution.

Contribution

It establishes a deep connection between the triviality of regular conditional distributions and the measurable selection property, providing a new perspective on their structure.

Findings

Regular conditional distributions trivialize the conditioning sigma-algebra under certain conditions.

Existence of measurable selections characterizes when regular conditional distributions are trivial.

The paper offers a new criterion linking triviality and measurable selections.

Abstract

We make the simple, and yet deep, observation that a regular conditional distribution (rcd) almost surely trivialises the conditioning $\sigma$-algebra if and only if there exists a "measurable selection" of regular conditional distributions for almost all the measures arising out of the original rcd.