Conditional Expectations in Banach spaces with RNP

Abstract

Let $X$ be a Banach space with RNP, $(\vO,\vS,μ)$ be a complete probability space and $\vG:\vO\to{cb(X)}$ (nonempty, closed convex and bounded subsets of $X$) be a multifunction. Assume that $\vX\subset\vS$ is a $\sigma$-algebra and the multimeasure $M$ defined by the Pettis integral of $\vG$ be such that the restriction of $M$ to $\vX$ is of $\sigma$-finite variation. Using a lifting, I prove the existence of an Effros measurable conditional expectation of $\vG$ and present its representation in terms of quasi-selections of $\vG$. I apply then the description to martingales of Pettis integrable multifunctions obtaining a scalarly equivalent martingale of measurable multifunctions with many martingale selections. In general the situation cannot be reduced to the separable space.