The one-way Fubini property and conditional independence: An equivalence result

A process defined by a continuum of random variables with non-degenerate idiosyncratic risk is not jointly measurable with respect to the usual product sigma-algebra. We show that the process is jointly measurable in a one-way Fubini extension of the product space if and only if there is a countably generated sigma-algebra given which the random variables are essentially pairwise conditionally independent, while their conditional distributions also satisfy a suitable joint measurability condition. Applications include: (i) new characterizations of essential pairwise independence and essential pairwise exchangeability; (ii) when a one-way Fubini extension exists, the need for the sample space to be saturated if there is an essentially random regular conditional distribution with respect to the usual product sigma-algebra. (c) 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

This study explores the conditions under which a process defined by a continuum of random variables with non-degenerate idiosyncratic risk is jointly measurable, and provides a specific sigma algebra condition. Applications of the research include new characterizations and results related to conditional independence.