# Commonotonicity and $L^1$ Random Variables

## TL;DR

This paper proves that in appropriate filtrations, any pair of integrable random variables can be represented as the conditional expectation of a pair of commonotone integrable variables, highlighting a fundamental property of their structure.

## Contribution

It establishes that every pair of integrable random variables can be realized as conditional expectations of commonotone variables within suitable filtrations, a novel structural insight.

## Key findings

- Any pair of integrable random variables can be represented as conditional expectations of commonotone variables.
- The result applies within suitable filtrations, broadening the understanding of dependence structures.
- It advances the theory of commonotonicity in the context of $L^1$ random variables.

## Abstract

It is proved that in suitable filtrations every pair of integrable random variables is the conditional expectation of a pair of commonotone integrable random variables.