Splitting of Liftings in Product Spaces II

TL;DR

Let $R\in P\circledast Q$ be a skew product of two probability spaces with a $Q$-disintegration $\{({\mathfrak A}_y,S_y):y\in Y\}$. The paper constructs liftings $\pi$ on ${\mathfrak A}\divideontimes{\mathfrak B}$ and fiberwise liftings $\sigma_y$ on $\widehat{S_y}$ so that $[\pi(f)]^y=\sigma_y([\pi(f)]^y)$ for all $f\in{\mathcal L}^{\infty}(\widehat{R_{\divideontimes}})$ and a.e. $y$; it proves the main theorems (notably mu25) to connect conditional expectations across the fibers with the liftings. Under absolute continuity assumptions $R\ll P\times Q$ (or $S_y\ll P|_{\mathfrak A_y}$), the liftings can be taken to take values in $\mathfrak A\widehat{\otimes}\mathfrak B$, and the work both generalizes and corrects earlier results such as Theorem 3.8 in mu25. Finally, the paper applies the framework to characterize stochastic processes possessing equivalent measurable versions and to derive measurability-preserving modifications.

Abstract

Let $(X, \mfA,P)$ and $(Y, \mfB,Q)$ be two probability spaces, $R$ be their skew product on the product $σ$-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 $σ$-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 $π$ on $\mcL^{\infty}(\wh{R_{\dd}})$ and liftings $σ_y$ on $\mcL^{\infty}(\wh{S_y})$ , $y\in Y$, such that \[ [π(f)]^y= σ_y\Bigl([π(f)]^y\Bigr) \qquad\mbox{for every} \quad y\in Y\quad\mbox{and every}\quad f\in\mcL^{\infty}(\wh{R_{\dd}}). \] 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}.

Theorems & Definitions (22)

• Definition 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Definition 2.4
• Remark 2.5