Conditional Expectations in Banach spaces with RNP

Let $X$ be a Banach space with RNP, $(\text{vO},\text{vS},\mu )$ be a complete probability space and $\text{vG}:\text{vO}\to cb(X)$ (nonempty, closed convex and bounded subsets of $X$) be a multifunction. Assume that $\text{vX}\subset \text{vS}$ is a $\sigma$-algebra and the multimeasure $M$ defined by the Pettis integral of $\text{vG}$ be such that the restriction of $M$ to $\text{vX}$ is of $\sigma$-finite variation. Using a lifting, I prove the existence of an Effros measurable conditional expectation of $\text{vG}$ and present its representation in terms of quasi-selections of $\text{vG}$. 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.