Splitting of Liftings in Product Spaces II

TL;DR

This paper extends the theory of liftings in product probability spaces, providing new results on the existence of compatible liftings and characterizing processes with equivalent measurable versions, generalizing previous work.

Contribution

It introduces a generalized framework for liftings in product spaces and corrects earlier theorems, with specific results for separable and absolutely continuous cases.

Findings

Existence of compatible liftings in product spaces.

Characterization of processes with equivalent measurable versions.

Generalization and correction of previous theorems.

Abstract

Let $(X, \mfA,P)$ and $(Y, \mfB,Q)$ be two probability spaces, $R$ be their skew product on the product $\sigma$-algebra $\mfA\otimes\mfB$ and $\{(\mfA_y,S_y)\colon y\in{Y}\}$ be a $Q$-disintegration of $R$. Then let $\mfA\dd\mfB$ be the $\sigma$-algebra generated $\mfA\otimes\mfB$ and by the family $\mcM:=\{E\subset{X\times{Y}}\colon \exists\;N\in\mfB_0\;\forall\;y\notin{N}\;\wh{S_y}(E^y)=0\}$ and $\wh{R_{\dd}}$ be the extension of $R$ such that $\mcM$ becomes the family of $\wh{R_*}$-zero sets ($\wh{S_y}$ is the completion of $S_y$ and $\mfB_0=\{B\in\mfB: Q(B)=0\}$). We prove that there exist a lifting $\pi$ on $\mcL^{\infty}(\wh{R_{\dd}})$ and liftings $\sigma_y$ on $\mcL^{\infty}(\wh{S_y})$ , $y\in Y$, such that \[ [\pi(f)]^y= \sigma_y\Bigl([\pi(f)]^y\Bigr) \qquad\mbox{for every} \quad y\in Y\quad\mbox{and every}\quad f\in\mcL^{\infty}(\wh{R_{\dd}}). \] In case of a separable $P$ and in case when $R\ll{P}\times{Q}$ a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a generalization and correction of \cite[Theorem 3.8]{mu25}.